An $H^{3}(G,{\mathbb T})$-valued index of symmetry-protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems

Abstract

We consider symmetry-protected topological phases with on-site finite group G symmetry $\beta $ for two-dimensional quantum spin systems. We show that they have $H^{3}(G,{\mathbb T})$-valued invariant.

1 Introduction

The notion of symmetry-protected topological phases was introduced by Gu and Wen [GW]. It is defined as follows: We consider the set of all Hamiltonians with some symmetry which have a unique gapped ground state in the bulk and can be smoothly deformed into a common trivial gapped Hamiltonian without closing the gap. We say two such Hamiltonians are equivalent if they can be smoothly deformed into each other without breaking the symmetry. We call an equivalence class of this classification a symmetry-protected topological (SPT) phase. Based on tensor network or quantum field theory analysis [CGLW, MGSC], it is conjectured that SPT phases with on-site finite group G symmetry for $\nu $-dimensional quantum spin systems have an $H^{\nu +1}(G,{\mathbb T})$-valued invariant. We proved that conjecture affirmatively in [O1] for $\nu =1$. In this paper, we show that the conjecture is also true for $\nu =2$.

We start by summarising the standard setup of $2$-dimensional quantum spin systems on the $2$-dimensional lattice ${\mathbb Z}^{2}$ [BR1, BR2]. We will freely use the basic notation in Section A. Throughout this paper, we fix some $2\le d\in \mathbb {N}$. We denote the algebra of $d\times d$ matrices by $\mathop {\mathrm {M}}\nolimits _{d}$.

For each subset $\Gamma $ of ${\mathbb Z}^2$, we denote the set of all finite subsets in $\Gamma $ by ${\mathfrak S}_{\Gamma }$. We introduce the Euclidean metric on ${\mathbb Z}^2$, inherited from ${\mathbb R}^2$. We denote by $\mathrm {d}(S_1,S_2)$ the distance between $S_1,S_2\subset {\mathbb Z}^2$. For a subset $\Gamma $ of ${\mathbb Z}^2$ and $r\in {\mathbb R}_{\ge 0}$, $\hat \Gamma {(r)}$ denotes all the points in ${\mathbb R}^{2}$ whose distance from $\Gamma $ is less than or equal to r. We also set $\Gamma (r):=\hat \Gamma {(r)}\cap {\mathbb Z}^{2}$. When we take a complement of $\Gamma \subset {\mathbb Z}^{2}$, it means $\Gamma ^{c}:={\mathbb Z}^{2}\setminus \Gamma $. For each $n\in {\mathbb N}$, we denote $[-n,n]^2\cap {\mathbb Z}^2$ by $\Lambda _n$.

For each $z\in {\mathbb Z}^2$, let ${\mathcal A}_{\{z\}}$ be an isomorphic copy of $\mathop {\mathrm {M}}\nolimits _{d}$, and for any finite subset $\Lambda \subset {\mathbb Z}^2$, we set ${\mathcal A}_{\Lambda } = \bigotimes _{z\in \Lambda }{\mathcal A}_{\{z\}}$. For finite $\Lambda $, the algebra ${\mathcal A}_{\Lambda } $ can be regarded as the set of all bounded operators acting on the Hilbert space $\bigotimes _{z\in \Lambda }{{\mathbb C}}^{d}$. We use this identification freely. If $\Lambda _1\subset \Lambda _2$, the algebra ${\mathcal A}_{\Lambda _1}$ is naturally embedded in ${\mathcal A}_{\Lambda _2}$ by tensoring its elements with the identity. For an infinite subset $\Gamma \subset {\mathbb Z}^{2}$, ${\mathcal A}_{\Gamma }$ is given as the inductive limit of the algebras ${\mathcal A}_{\Lambda }$ with $\Lambda \in {\mathfrak S}_{\Gamma }$. We call ${\mathcal A}_{\Gamma }$ the quantum spin system on $\Gamma $. For simplicity, we denote the $2$-dimensional quantum spin system ${\mathcal A}_{{\mathbb Z}^{2}}$ by ${\mathcal A}$. We also set ${\mathcal A}_{\text {loc}}:=\bigcup _{\Lambda \in {\mathfrak S}_{{\mathbb Z}^{2}}}{\mathcal A}_{\Lambda } $. For a subset $\Gamma _1$ of $\Gamma \subset {\mathbb Z}^{2}$, the algebra ${\mathcal A}_{\Gamma _1}$ can be regarded as a subalgebra of ${\mathcal A}_{\Gamma }$. With this identification, for $A\in {\mathcal A}_{\Gamma _1}$ we occasionally use the same symbol A to denote $A\otimes \mathbb I_{{\mathcal A}_{\Gamma \setminus \Gamma _1}}\in {\mathcal A}_\Gamma $. Similarly, an automorphism $\gamma $ on ${\mathcal A}_{\Gamma _1}$ can be naturally regarded as an automorphism $\gamma \otimes \mathop {\mathrm {id}}\nolimits _{{\mathcal A}_{\Gamma \setminus \Gamma _1}}$ on ${\mathcal A}_{\Gamma }$. We use this identification freely, and with a slight abuse of notation we occasionally denote $\gamma \otimes \mathop {\mathrm {id}}\nolimits _{{\mathcal A}_{\Gamma \setminus \Gamma _1}}$ by $\gamma $. Similarly, for disjoint $\Gamma _-,\Gamma _+\subset {\mathbb Z}^{2}$ and $\alpha _\pm \in \mathop {\mathrm {Aut}}\nolimits {\mathcal A}_{\Gamma \pm }$, we occasionally write $\alpha _-\otimes \alpha _+$ to denote $\left ( \alpha _-\otimes \mathop {\mathrm {id}}\nolimits _{\Gamma _-^c}\right ) \left ( \alpha _+\otimes \mathop {\mathrm {id}}\nolimits _{\Gamma _+^c}\right )$, under the given identification.

Throughout this paper we fix a finite group G and a unitary representation U on ${\mathbb C}^{d}$. Let $\Gamma \subset {\mathbb Z}^{2}$ be a nonempty subset. For each $g\in G$, there exists a unique automorphism $\beta ^{\Gamma }_g$ on ${\mathcal A}_{\Gamma }$ such that

(1.1)$$ \begin{align} \beta_{g}^{\Gamma}\left ( A\right )=\mathop{\mathrm{Ad}}\nolimits\left (\bigotimes_{I} U(g)\right )\left ( A\right ),\quad A\in{\mathcal A}_{I}, \ g\in G, \end{align} $$

for any finite subset I of $\Gamma $. We call the group homomorphism $\beta ^{\Gamma }: G\to \mathop {\mathrm {Aut}}\nolimits {\mathcal A}_{\Gamma }$ the on-site action of G on ${\mathcal A}_{\Gamma }$ given by U. For simplicity, we denote $\beta ^{{\mathbb Z}^{2}}_{g}$ by $\beta _{g}$.

A mathematical model of a quantum spin system is fully specified by its interaction $\Phi $. A uniformly bounded interaction on ${\mathcal A}$ is a map $\Phi : {\mathfrak S}_{{\mathbb Z}^{2}}\to {\mathcal A}_{\text {loc}}$ such that

(1.2)$$ \begin{align} \Phi(X)=\Phi(X)^*\in {\mathcal A}_{X},\quad X\in {\mathfrak S}_{{\mathbb Z}^{2}}, \end{align} $$

and

(1.3)$$ \begin{align} \sup_{X\in {\mathfrak S}_{{\mathbb Z}^{2}}}\left \lVert \Phi(X)\right \rVert<\infty. \end{align} $$

It is of finite range, with interaction length less than or equal to $R\in \mathbb {N}$ if $\Phi (X)=0$ for any $X\in {\mathfrak S}_{{\mathbb Z}^{2}}$ whose diameter is larger than R. An on-site interaction – that is, an interaction with $\Phi (X)=0$ unless X consists of a single point – is said to be trivial. An interaction $\Phi $ is $\beta $-invariant if $\beta _g(\Phi (X))=\Phi (X)$ for any $X\in {\mathfrak S}_{{\mathbb Z}^{2}}$. For a uniformly bounded and finite-range interaction $\Phi $ and $\Lambda \in {\mathfrak S}_{{\mathbb Z}^{2}}$, define the local Hamiltonian

(1.4)$$ \begin{align} \left ( H_\Phi\right )_\Lambda :=\sum_{X\subset\Lambda} \Phi(X) \end{align} $$

and denote the dynamics

(1.5)$$ \begin{align} \tau^{(\Lambda)\Phi}_t (A):=e^{it\left ( H_\Phi\right )_\Lambda}Ae^{-it\left ( H_\Phi\right )_\Lambda}, \quad t\in {\mathbb R},\ A\in{\mathcal A}. \end{align} $$

By the uniform boundedness and finite-rangeness of $\Phi $, for each $A\in {\mathcal A}$ the following limit exists

(1.6)$$ \begin{align} \lim_{\Lambda\to{\mathbb Z}^{\nu}} \tau^{(\Lambda),\Phi}_t\left ( A\right )=: \tau^{\Phi}_t\left ( A\right ),\quad t\in{\mathbb R}, \end{align} $$

which defines the dynamics $\tau ^{\Phi }$ on ${\mathcal A}$ [BR2]. For a uniformly bounded and finite-range interaction $\Phi $, a state $\varphi $ on ${\mathcal A}$ is called a $\tau ^{\Phi }$-ground state if the inequality $ -i\,\varphi (A^*{\delta _{\Phi }}(A))\ge 0 $ holds for any element A in the domain ${\mathcal D}({\delta _{\Phi }})$ of the generator ${\delta _\Phi }$. Let $\varphi $ be a $\tau ^\Phi $-ground state, with a Gelfand–Naimark–Segal (GNS) triple $\left ({\mathcal H}_\varphi ,\pi _\varphi ,\Omega _\varphi \right )$. Then there exists a unique positive operator $H_{\varphi ,\Phi }$ on ${\mathcal H}_\varphi $ such that $e^{itH_{\varphi ,\Phi }}\pi _\varphi (A)\Omega _\varphi =\pi _\varphi \left (\tau ^t_\Phi (A)\right )\Omega _\varphi $, for all $A\in {\mathcal A}$ and $t\in \mathbb R$. We call this $H_{\varphi ,\Phi }$ the bulk Hamiltonian associated with $\varphi $.

Definition 1.1. We say that an interaction $\Phi $ has a unique gapped ground state if (i) the $\tau ^\Phi $-ground state, which we denote as $\omega _{\Phi }$, is unique and (ii) there exists a $\gamma>0$ such that $\sigma \left (H_{\omega _{\Phi },\Phi }\right )\setminus \{0\}\subset [\gamma ,\infty )$, where $\sigma \left (H_{\omega _{\Phi },\Phi }\right )$ is the spectrum of $H_{\omega _{\Phi },\Phi }$. We denote by ${\mathcal P}_{UG} $ the set of all uniformly bounded finite-range interactions with unique gapped ground state. We denote by ${\mathcal P}_{UG\beta }$ the set of all uniformly bounded finite-range $\beta $-invariant interactions with unique gapped ground state.

In this paper we consider a classification problem of a subset of ${\mathcal P}_{UG\beta }$, models with short-range entanglement. To describe the models with short-range entanglement, we need to explain the classification problem of unique gapped ground-state phases without symmetry. For $\Gamma \subset {\mathbb Z}^{2}$, we denote by $\Pi _{\Gamma }:{\mathcal A}\to {\mathcal A}_{\Gamma }$ the conditional expectation with respect to the trace state. Let $f:(0,\infty )\to (0,\infty )$ be a continuous decreasing function with $\lim _{t\to \infty }f(t)=0$. For each $A\in {\mathcal A}$, define

(1.7)$$ \begin{align} \left \lVert A\right \rVert_f:=\left \lVert A\right \rVert + \sup_{N\in \mathbb{N}}\left (\frac{\left \lVert A-\Pi_{\Lambda_N} (A) \right \rVert} {f(N)} \right ). \end{align} $$

We denote by ${\mathcal D}_f$ the set of all $A\in {\mathcal A}$ such that $\left \lVert A\right \rVert _f<\infty $.

The classification of unique gapped ground-state phases ${\mathcal P}_{UG} $ without symmetry is the following:

Definition 1.2. Two interactions $\Phi _0,\Phi _1\in {\mathcal P}_{UG} $ are equivalent if there is a path of interactions $\Phi : [0,1]\to {\mathcal P}_{UG} $ satisfying the following:

1. 1. $\Phi (0)=\Phi _0$ and $\Phi (1)=\Phi _1$.

2. 2. For each $X\in {\mathfrak S}_{{\mathbb Z}^2}$, the map $[0,1]\ni s\to \Phi (X;s)\in {\mathcal A}_{X}$ is $C^1$. We denote by $\dot {\Phi }(X;s)$ the corresponding derivatives. The interaction obtained by differentiation is denoted by $\dot \Phi (s)$, for each $s\in [0,1]$.

3. 3. There is a number $R\in \mathbb {N}$ such that $X \in {\mathfrak S}_{{\mathbb Z}^2}$ and $\mathop {\mathrm {diam}}\nolimits {X}\ge R$ imply $\Phi (X;s)=0$, for all $s\in [0,1]$.

4. 4. Interactions are bounded as follows:

   (1.8)$$ \begin{align} C_b^{\Phi}:=\sup_{s\in\left[0,1\right]}\sup_{X\in {\mathfrak S}_{{\mathbb Z}^2}} \left ( \left \lVert \Phi\left ( X;s\right ) \right \rVert+\left \lVert \dot{\Phi} \left ( X;s\right ) \right \rVert \right )<\infty. \end{align} $$

5. 5. Setting

   (1.9)$$ \begin{align} b(\varepsilon):=\sup_{Z\in{\mathfrak S}_{{\mathbb Z}^2}} \sup_{s,s_0 \in\left[0,1\right],0<\left \lvert s-s_0\right \rvert<\varepsilon} \left \lVert \frac{\Phi(Z;s)-\Phi(Z;s_0)}{s-s_0}-\dot{\Phi}(Z;s_0) \right \rVert \end{align} $$
   for each $\varepsilon>0$, we have $\lim _{\varepsilon \to 0} b(\varepsilon )=0$.

6. 6. There exists a $\gamma>0$ such that $\sigma \left (H_{\omega _{\Phi (s)},\Phi (s)}\right )\setminus \{0\}\subset [\gamma ,\infty )$ for all $s\in [0,1]$, where $\sigma \left (H_{\omega _{\Phi (s)},\Phi (s)}\right )$ is the spectrum of $H_{\omega _{\Phi (s)},\Phi (s)}$.

7. 7. There exists $0<\eta <1$ satisfying the following: Set $\zeta (t):=e^{-t^{ \eta }}$. Then for each $A\in {\mathcal D}_\zeta $, $\omega _{\Phi (s)}(A)$ is differentiable with respect to s, and there is a constant $C_\zeta $ such that

   (1.10)$$ \begin{align} \left \lvert \frac{d}{ds}\omega_{\Phi(s)}(A) \right \rvert \le C_\zeta\left \lVert A\right \rVert_\zeta, \end{align} $$
   for any $A\in {\mathcal D}_\zeta $. (Recall definition (1.7)).

We write $\Phi _0\sim \Phi _1$ if $\Phi _0$ and $\Phi _1$ are equivalent. If $\Phi _0,\Phi _1\in {\mathcal P}_{UG\beta }$ and we can take the path in ${\mathcal P}_{UG\beta }$ – that is, so that $\beta _g\left ( \Phi (X;s)\right )=\Phi (X;s)$, $g\in G$, for all $s\in [0,1]$ – then we say $\Phi _0$ and $\Phi _1$ are $\beta $-equivalent and write $\Phi _0\sim _\beta \Phi _1$.

The reason we require these conditions is that we rely on the result in [MO]. The object we classify in this paper is the following:

Definition 1.3. Fix a trivial interaction $\Phi _0\in {\mathcal P}_{UG} $. We denote by ${\mathcal P}_{SL\beta }$ the set of all $\Phi \in {\mathcal P}_{UG\beta }$ such that $\Phi \sim \Phi _0$. Connected components of ${\mathcal P}_{SL\beta }$ with respect to $\sim _\beta $ are the SPT phases.

Because we have $\Phi _0\sim \tilde \Phi _0$ for any trivial $\Phi _0,\tilde \Phi _{0}\in {\mathcal P}_{UG} $, the set ${\mathcal P}_{SL\beta }$ does not depend on the choice of $\Phi _0$.

Remark 1.4. From the automorphic equivalence (Theorem 5.1), $\Phi \sim \Phi _0$ means that the ground state of $\Phi $ has a short-range entanglement. This is because the automorphisms in Theorem 5.1 can be regarded as a version of a quantum circuit with finite depth, which is regarded as a quantum circuit that does not create long-range entanglement [BL].

The main result of this paper is as follows:

Theorem 1.5. There is an $H^3(G,{\mathbb T})$-valued index on ${\mathcal P}_{SL\beta }$, which is an invariant of the classification $\sim _\beta $ of ${\mathcal P}_{SL\beta }$.

The paper is organised as follows. In Section 2, we define the $H^{3}(G,{\mathbb T})$-valued index for a class of states which are created from a fixed product state via ‘factorisable’ automorphisms, satisfying some additional condition. This additional condition is the existence of the set of automorphisms which (i) do not move the state and (ii) are almost like $\beta $-action restricted to the upper half-plane, except for some $1$-dimensional perturbation. In Section 3, we show that the existence of such set of automorphisms is guaranteed in a suitable situation. Furthermore, in Section 4 we show the stability of the index – that is, a suitably $\beta $-invariant automorphism does not change this index. Finally, in Section 5 we show our main theorem, Theorem 1.5, and that in our setting of Theorem 1.5, all the conditions required in Sections 2, 3 and 4 are satisfied. Although the index is defined in terms of GNS representations, in some good situation, we can calculate it without going through GNS representation; this is shown in Section 6. Reviews of this article can be found in [O3, O4].

2 The $H^{3}(G,{\mathbb T})$-valued index in $2$-dimensional systems

In this section, we associate an $H^{3}(G,{\mathbb T})$-index for some class of states. It will turn out later that this class includes SPT phases. For a nontrivial example of this index, see [O3]. It is also shown there that if a state is of product form of two states on half-planes, then our index is trivial. From the construction to follow, one can easily see that the group structure of $H^{3}(G,{\mathbb T})$, which is a simple pointwise multiplication, shows up when we tensor two systems.

2.1 An overview

We consider states of the form $\omega =\omega _{0}\circ \alpha $, where $\omega _{0}$ is a pure infinite tensor product state (see definition (2.18)) and $\alpha $ an automorphism satisfying some factorisation property (2.8). In equation (2.8), $\alpha _{L}, \alpha _{R}$ are automorphisms localised to the left and right infinite planes $H_{L}, H_{R}$, and $\Theta $ is localised in $\left ( C_{\theta }\right )^{c}$, where $C_{\theta }$ is defined by definition (2.2). We then have $\omega \simeq \left ( \omega _{L}\alpha _{L}\otimes \omega _{R}\alpha _{R}\right )\circ \Theta $ with pure states $\omega _{L}, \omega _{R}$ on the left and right infinite planes. We further assume that the effective excitation caused by $( \beta _{g}^{U})^{-1}$ (see formula (2.5)) on $\omega $ is localised around the x-axis, in the sense that for any $0<\theta <\frac \pi 2$, there are automorphisms $\eta _{g}^{L}, \eta _{g}^{R}$ localised in $C_{\theta }\cap H_{L}, C_{\theta }\cap H_{R}$ such that $\omega \circ ( \beta _{g}^{U})^{-1}$ is equivalent to $\omega \circ (\eta _{g}^{L}\otimes \eta _{g}^{R} )$. This corresponds to thinking of ${\mathcal T}(\theta , (\tilde \beta _g) )$ (definition (2.22)) and $\mathop {\mathrm {IG}}\nolimits (\omega ,\theta )$ (definition (2.24)). Setting $\gamma _{g}^{R}:=\eta _{g}^{R}\beta _{g}^{UR}, \gamma _{g}^{L}:=\eta _{g}^{L}\beta _{g}^{UL}$, with $\beta _{g}^{UR}, \beta _{g}^{UL}$ in formula (2.5), the condition given is $\omega \simeq \omega \circ ( \gamma _{g}^{L}\otimes \gamma _{g}^{R} )$. Repeated use of this formula gives us $\omega \simeq \omega \circ ( \gamma _{g}^{L}\gamma _{h}^{L}(\gamma _{gh}^{L} )^{-1} \otimes \gamma _{g}^{R}\gamma _{h}^{R} (\gamma _{gh}^{R} )^{-1} )$. Substituting the factorisation of $\omega $, we then have

$$ \begin{align*} \left ( \omega_{L}\alpha_{L}\otimes \omega_{R}\alpha_{R}\right )\circ\Theta \simeq \left ( \omega_{L}\alpha_{L}\otimes \omega_{R}\alpha_{R}\right )\circ\Theta \left ( \gamma_{g}^{L}\gamma_{h}^{L}\left (\gamma_{gh}^{L}\right )^{-1} \otimes \gamma_{g}^{R}\gamma_{h}^{R}\left (\gamma_{gh}^{R}\right )^{-1}\right ). \end{align*} $$

However, because conjugation by $\beta _g^U$ does not change the support of automorphisms, we see that this combination $\gamma _{g}^{R}\gamma _{h}^{R}(\gamma _{gh}^{R} )^{-1}$ is localised in $C_{\theta }\cap H_{R}$. As a result, $\gamma _{g}^{R}\gamma _{h}^{R}(\gamma _{gh}^{R})^{-1}$ – and also $\gamma _{g}^{L}\gamma _{h}^{L}(\gamma _{gh}^{L} )^{-1}$ – commutes with $\Theta $. Letting them commute, we obtain

$$ \begin{align*} \omega_{L}\alpha_{L}\otimes \omega_{R}\alpha_{R}\simeq \left ( \omega_{L}\alpha_{L}\otimes \omega_{R}\alpha_{R}\right )\circ \left ( \gamma_{g}^{L}\gamma_{h}^{L}\left (\gamma_{gh}^{L}\right )^{-1} \otimes \gamma_{g}^{R}\gamma_{h}^{R}\left (\gamma_{gh}^{R}\right )^{-1}\right ), \end{align*} $$

from which we can conclude $ \omega _{R}\alpha _{R}\simeq \omega _{R}\alpha _{R} \gamma _{g}^{R}\gamma _{h}^{R}(\gamma _{gh}^{R})^{-1}$. This means that $\alpha _{R} \gamma _{g}^{R}\gamma _{h}^{R}(\gamma _{gh}^{R})^{-1}\alpha _{R}^{-1}$ is implementable by some unitary $u(g,h)$ unitary in the GNS representation $\pi _{R}$ of $\omega _{R}$ (equation (2.19); see equation (2.27)). On the other hand, substituting the factorisation of $\omega $ to $\omega \simeq \omega \circ ( \gamma _{g}^{L}\otimes \gamma _{g}^{R} )$ implies

(2.1)$$ \begin{align} \left ( \omega_{L}\alpha_{L}\otimes \omega_{R}\alpha_{R}\right ) \simeq \left ( \omega_{L}\alpha_{L}\otimes \omega_{R}\alpha_{R}\right )\circ \Theta \circ\left ( \gamma_{g}^{L}\otimes \gamma_{g}^{R}\right )\circ \Theta^{-1}, \end{align} $$

from which we can derive the implementability of $\Theta \circ ( \gamma _{g}^{L}\otimes \gamma _{g}^{R} )\circ \Theta ^{-1}$ in the representation $\pi _{L}\alpha _{L}\otimes \pi _{R}\alpha _{R}$ by some unitary $W_{g}$ (see equation (2.26)). Using the definitions of $W_{g}$ and $u(g,h)$, we can see that they satisfy some nontrivial relation (2.52), with some $U(1)$-phase $c_{R}(g,h.k)$. In fact, this is quite a similar situation to that of cocycle actions [J]. As in [J], we can show that this $U(1)$-phase $c_{R}(g,h.k)$ is a $3$-cocycle and obtain an $H^{3}(G, {\mathbb T})$-index. The rest of this section is devoted to the proof that our index is independent of the choice of objects we introduced to define it. All of them follow from the fact that the difference of $W_{g}$ and $u(g,h)$ caused by the different choice of objects can be implemented by some unitary, and the proof is rather straightforward.

2.2 Definitions and the setting

For $0<\theta <\frac \pi 2$, a (double) cone $C_\theta $ is defined by

(2.2)$$ \begin{align} C_\theta:= \left\{ (x,y)\in{\mathbb Z}^2\mid \lvert y\rvert\le \tan \theta\cdot \lvert x\rvert \right\}. \end{align} $$

Note that it consists of the left part $x\le -1$ and the right part $0\le x$. For $0<\theta _1<\theta _2\le \frac \pi 2$, we use the notation ${\mathcal C}_{\left (\theta _1,\theta _2\right ]}:=C_{\theta _2}\setminus C_{\theta _1}$ and ${\mathcal C}_{\left [0,\theta _1\right ]}:=C_{\theta _1}$. Left, right, upper and lower half-planes are denoted by $H_L$, $H_R$, $H_U$ and $H_D$:

(2.3)$$ \begin{align} H_L & :=\left\{ (x,y)\in{\mathbb Z}^2\mid x\le -1\right\}, & H_R & :=\left\{ (x,y)\in{\mathbb Z}^2\mid 0\le x\right\}, \end{align} $$

(2.4)$$ \begin{align} H_U & :=\left\{ (x,y)\in{\mathbb Z}^2\mid 0\le y\right\}, & H_D & :=\left\{ (x,y)\in{\mathbb Z}^2\mid y\le -1\right\}. \end{align} $$

We use the notation

(2.5)$$ \begin{align} \beta_g:=\beta_g^{{\mathbb Z}^2}, \quad \beta_g^U:=\beta_g^{H_U},\quad \beta_g^{RU}:=\beta_g^{H_R\cap H_U},\quad \beta_g^{LU}:=\beta_g^{H_L\cap H_U}. \end{align} $$

For each subset S of ${\mathbb Z}^2$, we set

(2.6)$$ \begin{align} S_\sigma:=S\cap H_\sigma,\quad S_\zeta:=S\cap H_\zeta,\quad S_{\sigma\zeta}:=S\cap H_\sigma\cap H_\zeta, \quad \sigma=L,R,\ \zeta=U,D. \end{align} $$

We occasionally write ${\mathcal A}_{S,\sigma }, {\mathcal A}_{S,\zeta }, {\mathcal A}_{S,\sigma ,\zeta }$ to denote ${\mathcal A}_{S_{\sigma }}, {\mathcal A}_{S_{\zeta }}, {\mathcal A}_{S_{\sigma \zeta }}$. For an automorphism $\alpha $ on ${\mathcal A}$ and $0<\theta <\frac {\pi }2$, we denote by ${\mathfrak D}^{\theta }_\alpha $ a set of all triples $(\alpha _L,\alpha _R,\Theta )$ with

(2.7)$$ \begin{align} \alpha_L\in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{H_L}\right ),\quad \alpha_R\in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{H_R}\right ),\quad \Theta\in\mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{\left ( C_\theta\right )^c}\right ) \end{align} $$

decomposing $\alpha $ as

(2.8)$$ \begin{align} \alpha=({\textrm{{inner}}})\circ\left (\alpha_L\otimes\alpha_R\right )\circ\Theta. \end{align} $$

For $(\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{(\theta )}_\alpha $, we set

(2.9)$$ \begin{align} \alpha_0:=\alpha_L\otimes\alpha_R. \end{align} $$

The class of automorphisms which allow such decompositions for any directions is denoted by

(2.10)$$ \begin{align} \mathop{\mathrm{QAut}}\nolimits\left ( {\mathcal A}\right ) :=\left\{ \alpha\in \mathop{\mathrm{Aut}}\nolimits({\mathcal A})\mid {\mathfrak D}^{\theta}_\alpha\neq\emptyset \text{ for all } 0<\theta<\frac\pi 2 \right\}. \end{align} $$

Furthermore, for each

(2.11)$$ \begin{align} 0<\theta_{0.8}<\theta_1<\theta_{1.2}<\theta_{1.8}<\theta_2<\theta_{2.2}< \theta_{2.8}<\theta_3<\theta_{3.2}<\frac\pi 2, \end{align} $$

we consider decompositions of $\alpha \in \mathop {\mathrm {Aut}}\nolimits ({\mathcal A})$ such that

(2.12)$$ \begin{align} &\alpha=({\textrm{{inner}}})\circ\left ( \alpha_{\left[0,\theta_1\right]}\otimes\alpha_{\left(\theta_1,\theta_2\right]} \otimes \alpha_{\left(\theta_2,\theta_3\right]}\otimes \alpha_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \alpha_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ), \end{align} $$

with

(2.13)$$ \begin{align} \begin{aligned} \alpha_X & :=\bigotimes_{\sigma=L,R, \: \zeta=D,U} \alpha_{X,\sigma,\zeta}, & \alpha_{\left[0,\theta_1\right]} & :=\bigotimes_{\sigma=L,R}\alpha_{\left[0,\theta_{1}\right],\sigma}, & \alpha_{\left(\theta_3,\frac\pi 2\right]} & :=\bigotimes_{\zeta=D,U} \alpha_{\left(\theta_3,\frac\pi 2\right],\zeta}, \\ \alpha_{X,\sigma,\zeta} & \in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{X,\sigma,\zeta}}\right ), & \alpha_{X,\sigma} & :=\bigotimes_{\zeta=U,D}\alpha_{X,\sigma,\zeta}, & \alpha_{X,\zeta} & :=\bigotimes_{\sigma=L,R}\alpha_{X,\sigma,\zeta}, \\ \alpha_{\left[0,\theta_{1}\right],\sigma} & \in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{\left[0,\theta_{0}\right],\sigma}}\right ), & \alpha_{\left(\theta_3,\frac\pi 2\right],\zeta} &\in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{\left(\theta_3,\frac\pi 2\right],\zeta}}\right ), \end{aligned} \end{align} $$

for

(2.14)$$ \begin{align} X=(\theta_1,\theta_2], (\theta_2,\theta_3], (\theta_{0.8},\theta_{1.2}], (\theta_{1.8},\theta_{2.2}], (\theta_{2.8},\theta_{3.2}],\quad \sigma=L,R,\ \zeta=D,U. \end{align} $$

The class of automorphisms on ${\mathcal A}$ which allow such decompositions for any directions $\theta _{0.8}, \theta _1, \theta _{1.2}, \theta _{1.8},\theta _2,\theta _{2.2}, \theta _{2.8},\theta _3,\theta _{3.2}$ (satisfying formula (2.11)) is denoted by $\mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$. Note that $\mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})\subset \mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$. The set of all $\alpha \in \mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$ with each of $\alpha _{I}$ in the decompositions required to commute with $\beta _g^{U}$, $g\in G$, is denoted by $\mathop {\mathrm {GSQAut}}\nolimits ({\mathcal A})$:

(2.15)$$ \begin{align} \begin{split} &\mathop{\mathrm{GSQAut}}\nolimits({\mathcal A})\\ &:=\left\{ \alpha\in \mathop{\mathrm{SQAut}}\nolimits({\mathcal A})\middle\vert \begin{gathered} \text{for any } \theta_{0.8}, \theta_1, \theta_{1.2},\theta_{1.8},\theta_2,\theta_{2.2}, \theta_{2.8},\theta_3,\theta_{3.2}\text{ satisfying formula (2.11),}\\\text{there is a decomposition }(2.12), (2.13), (2.14) \text{ satisfying}\\ \alpha_{I}\circ\beta_g^{U}=\beta_g^{U}\circ\alpha_{I},\ g\in G,\\ \text{for all }I=[0,\theta_1],(\theta_1,\theta_2], (\theta_2,\theta_3], \left(\theta_3,\frac\pi 2\right], (\theta_{0.8}, \theta_{1.2}], (\theta_{1.8},\theta_{2.2}], (\theta_{2.8},\theta_{3.2}] \end{gathered} \right\}. \end{split} \end{align} $$

We also define

(2.16)$$ \begin{align} \mathop{\mathrm{HAut}}\nolimits\left ( {\mathcal A}\right ):= \left\{\alpha\in\mathop{\mathrm{Aut}}\nolimits({\mathcal A})\middle\vert \begin{gathered} \text{for any } 0<\theta<\frac\pi 2, \text{ there exist }\alpha_{\sigma}\in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{{\left ( C_{\theta}\right )_\sigma}}\right ),\ \sigma=L,R,\\ \text{such that } \alpha=({\textrm{{inner}}})\circ\left ( \alpha_{L}\otimes \alpha_{R}\right ) \end{gathered} \right\}. \end{align} $$

In Section 5, we will see that quasilocal automorphisms corresponding to paths in symmetric gapped phases belong to the following set:

(2.17)$$ \begin{align} \mathop{\mathrm{GUQAut}}\nolimits\left ({\mathcal A}\right ) :=\left\{ \gamma\in\mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}\right )\middle\vert \begin{gathered} \text{there are } \gamma_{H}\in \mathop{\mathrm{HAut}}\nolimits({\mathcal A}), \gamma_{C}\in \mathop{\mathrm{GSQAut}}\nolimits({\mathcal A}),\\ \text{such that } \gamma=\gamma_{C}\circ\gamma_{H} \end{gathered} \right\}. \end{align} $$

We fix a reference state $\omega _0$ as follows: We fix a unit vector ${\xi _x}\in {\mathbb C}^d$ and let $\rho _{\xi _x}$ be the vector state on $\mathop {\mathrm {M}}\nolimits _d$ given by ${\xi _x}$, for each $x\in {\mathbb Z}^2$. Then our reference state $\omega _0$ is given by

(2.18)$$ \begin{align} \omega_0:=\bigotimes_{x\in {\mathbb Z}^2} \rho_{\xi_x}. \end{align} $$

Throughout this section this $\omega _0$ is fixed. Let $({\mathcal H}_0,\pi _0,\Omega _0)$ be a GNS triple of $\omega _0$. Because of the product structure of $\omega _0$, it is decomposed as

(2.19)$$ \begin{align} {\mathcal H}_0={\mathcal H}_L\otimes{\mathcal H}_R,\quad \pi_0=\pi_L\otimes\pi_R,\quad \Omega_0=\Omega_L\otimes\Omega_R, \end{align} $$

where $({\mathcal H}_\sigma ,\pi _\sigma ,\Omega _\sigma )$ is a GNS triple of $\omega _\sigma :=\omega _0\rvert _{{\mathcal A}_{H_\sigma }}$ for $\sigma =L,R$. As $\omega _0\rvert _{{\mathcal A}_{H_\sigma }}$ is pure, $\pi _\sigma $ is irreducible. What we consider in this section is the set of states created via elements in $\mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$ from our reference state $\omega _0$:

(2.20)$$ \begin{align} \mathop{\mathcal{SL}}\nolimits:=\left\{\omega_0\circ\alpha\mid \alpha\in \mathop{\mathrm{QAut}}\nolimits({\mathcal A}) \right\}. \end{align} $$

Because any pure product states can be transformed to each other via an automorphism of product form $\tilde \alpha =\bigotimes _{x\in {\mathbb Z}^{2}}\tilde \alpha _{x}$, and $\tilde \alpha \alpha $ belongs to $\mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$ for any $\alpha \in \mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$, $\mathop {\mathcal {SL}}\nolimits $ does not depend on the choice of $\omega _0$. For each $\omega \in \mathop {\mathcal {SL}}\nolimits $, we set

(2.21)$$ \begin{align} \mathop{\mathrm{EAut}}\nolimits(\omega):= \left\{ \alpha\in \mathop{\mathrm{QAut}}\nolimits({\mathcal A})\mid \omega=\omega_0\circ\alpha \right\}. \end{align} $$

By the definition of $\mathop {\mathcal {SL}}\nolimits $, $\mathop {\mathrm {EAut}}\nolimits (\omega )$ is not empty.

For $0<\theta <\frac \pi 2$ and a set of automorphisms $\left (\tilde \beta _g\right )_{g\in G}\subset \mathop {\mathrm {Aut}}\nolimits ({{\mathcal A}})$, we introduce a set

(2.22)$$ \begin{align} {\mathcal T}\left(\theta, (\tilde\beta_g )\right) :=\left\{ (\eta_{g}^\sigma )_{g\in G,\, \sigma=L,R } \middle\vert \begin{gathered} \eta_g^\sigma\in\mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{\left ( C_\theta\right )_\sigma}\right ), \\ \tilde\beta_g=({\textrm{{inner}}})\circ\left ( \eta_g^L\otimes\eta_g^R\right )\circ\beta_g^U,\\ \text{for all}\;g\in G,\ \sigma=L,R \end{gathered} \right\}. \end{align} $$

In a word, it is a set of decompositions of $\tilde \beta _g\circ ( \beta _g^{U})^{-1}$ into tensors of $\mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{( C_\theta )_L} ), \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{( C_\theta )_R} )$ modulo inner automorphisms. For $(\eta _{g}^\sigma )_{g\in G,\, \sigma =L,R }\in {\mathcal T}(\theta , (\tilde \beta _g ))$, we set

(2.23)$$ \begin{align} \eta_g:=\eta_g^L\otimes\eta_g^R,\quad g\in G. \end{align} $$

The following set of automorphisms is the key ingredient for the definition of our index: For $\omega \in \mathop {\mathcal {SL}}\nolimits $ and $0<\theta <\frac \pi 2$, we set

(2.24)$$ \begin{align} \mathop{\mathrm{IG}}\nolimits\left (\omega,\theta\right ) :=\left\{ (\tilde\beta_g )_{g\in G}\in\mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}\right )^{\times G} \middle\vert \begin{gathered} \omega\circ\tilde\beta_g=\omega \text{ for all } g\in G\\ \text{and }{\mathcal T}\left(\theta, (\tilde\beta_g )\right)\neq\emptyset\end{gathered} \right\}. \end{align} $$

We also set

(2.25)$$ \begin{align} \mathop{\mathrm{IG}}\nolimits\left (\omega\right ):=\cup_{0<\theta<\frac \pi 2}\mathop{\mathrm{IG}}\nolimits\left (\omega,\theta\right ). \end{align} $$

In this section we associate some third cohomology $h(\omega )$ for each $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $.

2.3 Derivation of elements in $Z^3(G,{\mathbb T})$

In this subsection, we derive $3$-cocycles out of $\omega , \alpha , \theta , (\tilde \beta _g ), (\eta _{g}^\sigma ), (\alpha _L,\alpha _R,\Theta )$.

Lemma 2.1. Set $\omega \in \mathop {\mathcal {SL}}\nolimits , \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega ), 0<\theta <\frac \pi 2, (\tilde \beta _g)\in \mathop {\mathrm {IG}}\nolimits \left (\omega ,\theta \right ), (\eta _{g}^\sigma )\in {\mathcal T}(\theta , (\tilde \beta _g)), (\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{\theta }_\alpha $. Then the following hold:

1. (i) There are unitaries $W_g$, $g\in G$, on ${\mathcal H}_0$ such that

   (2.26)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g)\circ\pi_0 =\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1},\quad g\in G, \end{align} $$
   with the notation of definitions (2.9) and (2.23).

2. (ii) There exists a unitary $u_\sigma (g,h)$ on ${\mathcal H}_{\sigma }$, for each $\sigma =L,R$ and for $g,h\in G$, such that

   (2.27)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_\sigma(g,h)\right )\circ\pi_\sigma =\pi_\sigma\circ\alpha_\sigma\circ\eta_g^\sigma\beta_g^{\sigma U} \eta_h^\sigma\left (\beta_g^{\sigma U}\right )^{-1}\left ( \eta_{gh}^\sigma\right )^{-1} \circ\alpha_\sigma^{-1} \end{align} $$
   and
   (2.28)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_L(g,h)\otimes u_R(g,h)\right )\pi_0 =\pi_0\circ\alpha_0\circ\eta_g\beta_g^U\eta_h\left ( \beta_g^U\right )^{-1}\left (\eta_{gh}\right )^{-1} \circ\alpha_0^{-1}. \end{align} $$
   Furthermore, $u_{\sigma }(g,h)$ commutes with any element of $ \pi _\sigma \circ \alpha _\sigma \left ( {\mathcal A}_{\left ( \left ( C_\theta \right )^c\right )_\sigma }\right )$.

Definition 2.2. For $\omega \in \mathop {\mathcal {SL}}\nolimits , \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega ), 0<\theta <\frac \pi 2, (\tilde \beta _g )\in \mathop {\mathrm {IG}}\nolimits (\omega ,\theta ), (\eta _{g}^\sigma)_{g\in G,\, \sigma =L,R }\in {\mathcal T}\left (\theta , \left (\tilde \beta _g\right )\right ), (\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{\theta }_\alpha $, we denote by

(2.29)$$ \begin{align} \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ) \end{align} $$

the set of $( (W_g)_{g\in G}, (u_\sigma (g,h))_{g,h\in G,\,\sigma =L,R} )$ with $W_{g}\in {\mathcal U}({\mathcal H}_{0} )$ and $u_\sigma (g,h)\in {\mathcal U}\left ({\mathcal H}_{\sigma }\right )$ satisfying

(2.30)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g)\circ\pi_0 & =\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1},\quad g\in G, \end{align} $$

and

(2.31)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_{\sigma}(g,h)\right )\circ\pi_{\sigma} & =\pi_{\sigma}\circ\alpha_{\sigma}\circ\eta_g^{\sigma}\beta_g^{{\sigma} U} \eta_h^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}( \eta_{gh}^{\sigma})^{-1} \circ\alpha_{\sigma}^{-1},\quad g,h\in G, \ \sigma=L,R. \end{align} $$

(Here we used the notation of definition (2.9) and (2.23).) By Lemma 2.1, it is nonempty.

Proof. For a GNS triple $({\mathcal H}_0,\pi _0\circ \alpha ,\Omega _0)$ of $\omega =\omega _0\circ \alpha $, there are unitaries $\tilde W_g$ on ${\mathcal H}_0$ such that

(2.32)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left (\tilde W_g\right )\circ\pi_0\circ\alpha =\pi_0\circ\alpha\circ\tilde\beta_g,\quad g\in G, \end{align} $$

because $\omega \circ \tilde \beta _g=\omega $.

Because $(\eta _{g}^\sigma )_{g\in G,\, \sigma =L,R }\in {\mathcal T}(\theta , (\tilde \beta _g ) )$ and $(\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{\theta }_\alpha $, there are unitaries $v_g, V\in {\mathcal U}\left ({\mathcal A}\right )$ such that

(2.33)$$ \begin{align} \tilde\beta_g=\mathop{\mathrm{Ad}}\nolimits\left ( v_g\right )\circ\left ( \eta_g^L\otimes\eta_g^R\right )\circ\beta_g^U,\quad \alpha=\mathop{\mathrm{Ad}}\nolimits V\circ\alpha_0\circ\Theta. \end{align} $$

Substituting these, we have

(2.34)$$ \begin{align} \begin{split} \mathop{\mathrm{Ad}}\nolimits \left ( \tilde W_g\pi_0(V)\right )\pi_0\circ\alpha_0\circ\Theta & =\pi_0\circ\alpha\tilde\beta_g =\pi_0\circ\alpha\circ\mathop{\mathrm{Ad}}\nolimits\left(v_g\right)\circ \eta_g\beta_g^U\\ &=\mathop{\mathrm{Ad}}\nolimits\left ( \left ( \pi_0\circ\alpha(v_g)\right )\pi_0(V) \right )\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U. \end{split} \end{align} $$

Therefore, setting $W_g:=\pi _0(V)^*( \pi _0\circ \alpha (v_g^*))\tilde W_g\pi _0(V)\in {\mathcal U}({\mathcal H}_0)$, we obtain equation (2.26).

Using equation (2.26), we have

(2.35)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( W_gW_h W_{gh}^*\right ) \pi_0 =\pi_0\circ\alpha_0\circ\Theta \circ\eta_g\beta_g^U\eta_h\left ( \beta_g^{U}\right )^{-1} \eta_{gh}^{-1}\Theta^{-1}\alpha_0^{-1}. \end{align} $$

Note that because conjugation by $\beta _g^U$ does not change the support of automorphisms, $\eta _g\beta _g^U\eta _h( \beta _g^{U} )^{-1} \eta _{gh}^{-1}$ belongs to $\mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{C_\theta } )$. On the other hand, $\Theta $ belongs to $\mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{ ( C_\theta )^{c}} )$. Therefore, they commute and we obtain

(2.36)$$ \begin{align} \begin{split} \mathop{\mathrm{Ad}}\nolimits\left ( W_gW_h W_{gh}^*\right ) \pi_0 &= \text{equation }(2.35)=\pi_0\circ\alpha_0\circ\eta_g\beta_g^U\eta_h\left ( \beta_g^{U}\right )^{-1} \eta_{gh}^{-1}\alpha_0^{-1}\\ &=\bigotimes_{\sigma=L,R} \pi_\sigma\circ\alpha_\sigma\circ\eta_g^\sigma\beta_g^{\sigma U} \eta_h^\sigma\left (\beta_g^{\sigma U}\right )^{-1}\left ( \eta_{gh}^\sigma\right )^{-1} \circ\alpha_\sigma^{-1}. \end{split} \end{align} $$

From this and the irreducibility of $\pi _R$, we see that $\mathop {\mathrm {Ad}}\nolimits ( W_gW_h W_{gh}^* )$ gives rise to a $*$-isomorphism $\tau $ on ${\mathcal B}({\mathcal H}_R)$. It is implemented by some unitary $u_R(g,h)$ on ${\mathcal H}_R$ by the Wigner theorem, and we obtain

(2.37)$$ \begin{align} \begin{split} \mathbb I_{{\mathcal H}_L}\otimes \left ( \mathop{\mathrm{Ad}}\nolimits\left ( u_R(g,h) \right )\circ\pi_R(A)\right ) &=\mathbb I_{{\mathcal H}_L}\otimes \tau\left ( \pi_R(A)\right ) =\mathop{\mathrm{Ad}}\nolimits\left ( W_gW_h W_{gh}^*\right )\left ( \mathbb I_{{\mathcal H}_L}\otimes \pi_R(A)\right )\\ &=\mathbb I_{{\mathcal H}_L}\otimes \pi_R\circ\alpha_R\circ\eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1} \circ\alpha_R^{-1}(A), \end{split} \end{align} $$

for any $A\in {\mathcal A}_{H_{R}}$. Hence we obtain equation (2.27) for $\sigma =R$.

To see that $u_R(g,h)$ belongs to $\left ( \pi _R\circ \alpha _R\left ( {\mathcal A}_{\left ( \left ( C_\theta \right )^c\right )_R}\right )\right )'$, set $A\in {\mathcal A}_{\left ( \left ( C_\theta \right )^c\right )_R}$. Then because $\eta _g^R\beta _g^{R U} \eta _h^R (\beta _g^{R U} )^{-1} ( \eta _{gh}^R )^{-1}$ belongs to $\mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{\left ( C_\theta \right )_R}\right )$, we have

(2.38)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_R(g,h)\right )\pi_R\left ( \alpha_R(A)\right ) =\pi_R\alpha_R \eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\alpha_R^{-1}\alpha_R(A) =\pi_R\alpha_R(A). \end{align} $$

This proves that $u_R(g,h)$ belongs to $\left ( \pi _R\circ \alpha _R\left ( {\mathcal A}_{\left ( \left ( C_\theta \right )^c\right )_R}\right )\right )'$. An analogous statement for $u_L(g,h)$ can be shown exactly the same way. The last statement of (ii), equation (2.28), is trivial from equation (2.27).

Lemma 2.3. Set $\omega \in \mathop {\mathcal {SL}}\nolimits , \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega ), 0<\theta <\frac \pi 2, \left (\tilde \beta _g\right )\in \mathop {\mathrm {IG}}\nolimits \left (\omega ,\theta \right ), (\eta _{g}^\sigma )\in {\mathcal T}(\theta , (\tilde \beta _g)), (\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{\theta }_\alpha $. Let $( (W_g), (u_R(g,h))) $ be an element of $\mathop {\mathrm {IP}}\nolimits ( \omega , \alpha , \theta , (\tilde \beta _g ), (\eta _{g}^\sigma ), (\alpha _L,\alpha _R,\Theta ) )$.

Then the following hold:

1. (i) For any $g,h,k\in G$,

   (2.39)$$ \begin{align} &\mathop{\mathrm{Ad}}\nolimits\left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right )\circ\pi_0 \nonumber \\ &\quad =\pi_0\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_{L}}}\otimes \alpha_R\eta_g^R\beta_g^{RU} \left ( \eta_h^R\beta_h^{R U} \eta_k^R\left (\beta_h^{R U}\right )^{-1}\left ( \eta_{hk}^R\right )^{-1} \right ) \left ( \eta_g^R\beta_g^{RU}\right )^{-1} \alpha_R^{-1} \right ). \end{align} $$

2. (ii) For any $g,h\in G$,

   (2.40)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( \left ( u_L(g,h)\otimes u_R(g,h)\right ) W_{gh}\right ) =\mathop{\mathrm{Ad}}\nolimits \left ( W_gW_h\right ) \end{align} $$
   on ${\mathcal B}({\mathcal H}_{0})$.

3. (iii) For any $g,h,k\in G$,

   (2.41)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) \in{\mathbb C}\mathbb I_{{\mathcal H}_L}\otimes{\mathcal B}({\mathcal H}_R). \end{align} $$

4. (iv) For any $g,h,k,f\in G$,

   (2.42)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( W_g W_h\right )\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ) =\left ( \mathop{\mathrm{Ad}}\nolimits\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h)\right ) W_{gh}\right ) \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ). \end{align} $$

Proof. We use the notation from definitions (2.9) and (2.23).

(i) Substituting equations (2.30) and (2.31), we have

(2.43)$$ \begin{align} \begin{split} \mathop{\mathrm{Ad}}\nolimits & \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right )\circ\pi_0 \\ &=\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1}\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_L}}\otimes \alpha_R\circ\eta_h^R\beta_h^{R U} \eta_k^R\left (\beta_h^{R U}\right )^{-1}\left ( \eta_{hk}^R\right )^{-1} \circ\alpha_R^{-1}\right )\\ &\quad \circ\alpha_0\circ\Theta\circ\left ( \eta_g\beta_g^U\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1} \\ &=\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_L}}\otimes \eta_h^R\beta_h^{R U} \eta_k^R\left (\beta_h^{R U}\right )^{-1}\left ( \eta_{hk}^R\right )^{-1} \right ) \circ\Theta\circ\left ( \eta_g\beta_g^U\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}. \end{split} \end{align} $$

Because $\eta _h^R\beta _h^{R U} \eta _k^R(\beta _h^{R U} )^{-1}( \eta _{hk}^R)^{-1}$ belongs to $\mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{\left ( C_\theta \right )_R}\right )$, it commutes with $\Theta \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{\left ( C_\theta \right )^{c}}\right )$. Hence we obtain

(2.44)$$ \begin{align} & \text{equation }(2.43)\notag\\ & \quad = \pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ \left ( \mathbb I_{{\mathcal A}_{H_L}}\otimes \eta_h^R\beta_h^{R U} \eta_k^R\left (\beta_h^{R U}\right )^{-1}\left ( \eta_{hk}^R\right )^{-1} \right ) \circ\left ( \eta_g\beta_g^U\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}\notag\\ & \quad =\pi_0\circ\alpha_0\circ\Theta\circ \left ( \mathbb I_{{\mathcal A}_{H_L}}\otimes \eta_g^R\beta_g^{RU}\circ\eta_h^R\beta_h^{R U} \eta_k^R\left (\beta_h^{R U}\right )^{-1}\left ( \eta_{hk}^R\right )^{-1}\circ \left ( \eta_g^R\beta_g^{RU}\right )^{-1} \right ) \circ\Theta^{-1}\circ\alpha_0^{-1}. \end{align} $$

Again, the term in parentheses in the last line is localised at $\left ( C_\theta \right )_R$, and it commutes with $\Theta $. Therefore, we have

(2.45)$$ \begin{align} &\mathop{\mathrm{Ad}}\nolimits\left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right )\circ\pi_0\nonumber\\ &\qquad=\pi_0\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_L}}\otimes \alpha_R\circ\eta_g^R\beta_g^{RU}\circ\eta_h^R\beta_h^{R U} \eta_k^R\left (\beta_h^{R U}\right )^{-1}\left ( \eta_{hk}^R\right )^{-1}\circ \left ( \eta_g^R\beta_g^{RU}\right )^{-1} \circ\alpha_R^{-1}\right ). \end{align} $$

(ii) Again by equations (2.30) and (2.31), we have

(2.46)$$ \begin{align} \begin{split} \mathop{\mathrm{Ad}}\nolimits & \left ( \left ( u_L(g,h)\otimes u_R(g,h)\right ) W_{gh}\right )\circ\pi_0 =\pi_0\circ\alpha_0\circ\eta_g\beta_g^U\eta_h\left ( \beta_g^U\right )^{-1}\left (\eta_{gh}\right )^{-1} \circ\Theta\circ\eta_{gh}\beta_{gh}^U\circ\Theta^{-1}\circ\alpha_0^{-1} \\ &=\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\eta_h\left ( \beta_g^U\right )^{-1}\left (\eta_{gh}\right )^{-1} \circ\eta_{gh}\beta_{gh}^U\circ\Theta^{-1}\circ\alpha_0^{-1}\\ &=\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\eta_h\beta_{h}^U\circ\Theta^{-1}\circ\alpha_0^{-1}\\ &=\mathop{\mathrm{Ad}}\nolimits \left ( W_gW_h\right )\circ\pi_0. \end{split} \end{align} $$

Here, for the second equality we again used the commutativity of $\eta $s and $\Theta $, due to their disjoint support. Because $\pi _0$ is irreducible, we obtain equation (2.40).

(iii) For any $A\in {\mathcal A}_{H_L}$, we have

(2.47)$$ \begin{align} \Theta^{-1}\circ\alpha_0^{-1}\left ( A\otimes \mathbb I_{{\mathcal A}_{H_R}}\right ) =\Theta^{-1}\circ\left ( \alpha_L^{-1}(A)\otimes \mathbb I_{{\mathcal A}_{H_R}}\right ) \in \Theta^{-1}\left ({\mathcal A}_{H_L}\otimes {\mathbb C}\mathbb I_{{\mathcal A}_{H_R}}\right ) \subset {\mathcal A}_{H_L\cup \left ( C_\theta^c\right )_R }, \end{align} $$

because $\Theta \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{\left ( C_\theta \right )^{c}}\right )$. Therefore, $\eta _g^R\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{\left ( C_\theta \right )_R}\right )$ acts trivially on it and we have

(2.48)$$ \begin{align} \left ( \beta_g^U\right )^{-1}\left ( \eta_g\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}\left ( A\otimes \mathbb I_{{\mathcal A}_{H_R}}\right ) \in {\mathcal A}_{H_L\cup \left ( C_\theta^c\right )_R }. \end{align} $$

As $\Theta $ preserves ${\mathcal A}_{H_L\cup \left ( C_\theta ^c\right )_R }$,

(2.49)$$ \begin{align} \Theta\circ\left ( \beta_g^U\right )^{-1}\left ( \eta_g\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}\left ( A\otimes \mathbb I_{{\mathcal A}_{H_R}}\right ) \end{align} $$

also belongs to ${\mathcal A}_{H_L\cup \left ( C_\theta ^c\right )_R }$. As a result,

(2.50)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( W_g^*\right )\left ( \pi_L(A)\otimes \mathbb I_{{\mathcal H}_R}\right ) =\pi_0\circ\alpha_0\circ\Theta\circ\left ( \beta_g^U\right )^{-1}\left ( \eta_g\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}\left ( A\otimes \mathbb I_{{\mathcal A}_{H_R}}\right ) \end{align} $$

belongs to $\pi _L({\mathcal A}_{H_L})\otimes \pi _R\circ \alpha _R( {\mathcal A}_{( C_\theta ^c)_R } )$, and hence commutes with $\mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)$. Hence $\mathop {\mathrm {Ad}}\nolimits (W_g)\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k) \right )$ commutes with any elements in $\pi _L({\mathcal A}_L)\otimes {\mathbb C}\mathbb I_{{\mathcal H}_R}$. Because $\pi _L$ is irreducible, $\mathop {\mathrm {Ad}}\nolimits (W_g)\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k) \right )$ belongs to ${\mathbb C}\mathbb I_{{\mathcal H}_L}\otimes {\mathcal B}({\mathcal H}_R)$.

(iv) By (iii), $\mathop {\mathrm {Ad}}\nolimits \left ( W_{gh}\right )\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right )$ belongs to ${\mathbb C}\mathbb I_{{\mathcal H}_L}\otimes {\mathcal B}({\mathcal H}_R)$. Therefore, from (ii), we have

(2.51)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( W_g W_h\right )\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ) & =\mathop{\mathrm{Ad}}\nolimits\left ( \left ( u_L(g,h)\otimes u_R(g,h)\right ) W_{gh}\right )\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right )\notag\\ &=\mathop{\mathrm{Ad}}\nolimits\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h)\right ) W_{gh}\right )\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ), \end{align} $$

obtaining (iv).

With this preparation, we may obtain some element of $Z^3(G,{\mathbb T})$ from $\left ((W_g), (u_\sigma (g,h))\right )$.

Lemma 2.4. Set $\omega \in \mathop {\mathcal {SL}}\nolimits , \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega ), 0<\theta <\frac \pi 2, (\tilde \beta _g)\in \mathop {\mathrm {IG}}\nolimits (\omega ,\theta ), (\eta _{g}^\sigma )\in {\mathcal T}(\theta , (\tilde \beta _g)), (\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{\theta }_\alpha $. Let $( (W_g), (u_\sigma (g,h)))$ be an element of $ \mathop {\mathrm {IP}}\nolimits ( \omega , \alpha , \theta , (\tilde \beta _g ), (\eta _{g}^\sigma ), (\alpha _L,\alpha _R,\Theta ) )$. Then there is a $c_R\in Z^3(G,{\mathbb T})$ such that

(2.52)$$ \begin{align} \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h) u_R(gh,k) =c_R(g,h,k) \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)\right ), \end{align} $$

for all $g,h,k\in G$.

Definition 2.5. We denote this $3$-cocycle $c_R$ in Lemma 2.4 by

(2.53)$$ \begin{align} c_R\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ) \end{align} $$

and its cohomology class by

(2.54)$$ \begin{align} &h^{(1)} \left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right )\nonumber\\ &\qquad\qquad\qquad:= \left [ c_R\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ) \right ]_{H^3\left(G,{\mathbb T}\right)}. \end{align} $$

Proof. First we prove that there is a number $c_R(g,h,k)\in {\mathbb T}$ satisfying equation (2.52). From equation (2.31), we have

(2.55)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h) u_R(gh,k) \right )\pi_0 =\pi_L\otimes \pi_R \circ \alpha_R\circ \left ( \eta_g^R\beta_g^{RU}\right ) \left ( \eta_h^R\beta_h^{RU}\right ) \left ( \eta_k^R\beta_k^{RU}\right ) \left ( \eta_{ghk}^R\beta_{ghk}^{RU}\right )^{-1} \alpha_R^{-1}. \end{align} $$

On the other hand, using Lemma 2.3(i), we have that

(2.56)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)\right )\right ) \pi_0 \end{align} $$

is also equal to the right-hand side of equation (2.55). Because $\pi _0$ is irreducible, this means that there is a number $c_R(g,h,k)\in {\mathbb T}$ satisfying equation (2.52).

Now let us check that this $c_R$ is a $3$-cocycle. For any $g,h,k,f\in G$, by repeated use of equation (2.52), we get

$$\begin{align*} \mathbb I_{{\mathcal H}_L} & \otimes u_R(g,h) u_R(gh,k)u_R(ghk,f) =\left [ \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h) u_R(gh,k)\right ]\cdot\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(ghk,f)\right ) \end{align*}$$

(2.57)$$ \begin{align} \nonumber\\[-24pt] & =\left ( c_R(g,h,k) \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)\right )\right )\cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(ghk,f)\right ) \nonumber \\ & =\left ( c_R(g,h,k) \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k) \right ) W_g^*\right )\right )\cdot \left [ \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)u_R(ghk,f)\right ] \nonumber\\ & =\left ( c_R(g,h,k) \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k) \right ) W_g^*\right )\right ) \nonumber\\ &\quad \cdot \left ( c_R(g,hk,f) \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(hk,f)\right ) W_g^*\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hkf)\right ) \right )\nonumber\\ &= c_R(g,h,k) c_R(g,hk,f) \left ( W_g\left [ \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k) u_R(hk,f)\right ] W_g^*\right )\cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hkf)\right ) \nonumber\\ &=c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f) \nonumber\\ &\quad W_g\left ( W_h \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ) W_h^*\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,kf)\right )\right ) W_g^*\cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hkf)\right )\nonumber\\ &=c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f)\nonumber\\ &\quad \cdot W_gW_h \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ) W_h^*W_g^*\cdot \left [\left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,kf)\right ) W_g^*\right ) \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hkf)\right ] \nonumber\\ & =c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f) \overline{c(g,h,kf)} \cdot \left\{W_gW_h \left (\mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right) \left.W_h^*W_g^*\right)\right\} \nonumber\\ &\quad \cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}(g,h)u_{R}(gh,kf) \right ). \end{align} $$

Here and in the following, we apply equation (2.52) for terms in $\left [\cdot \right ]$ to get the succeeding equality. Applying Lemma 2.3(iv) to the $\{\cdot \}$ part of equation (2.57), we obtain

(2.58)$$ \begin{align} \begin{split} \text{equation }(2.57) &=c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f)\overline{c(g,h,kf)}\\ &\quad \left ( \mathop{\mathrm{Ad}}\nolimits\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h)\right ) W_{gh}\right ) \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}(g,h)u_{R}(gh,kf) \right )\\ &=c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f)\overline{c(g,h,kf)}\\ &\quad \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h)\right ) \left [ W_{gh}\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(k,f)\right ) W_{gh}^{*} \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}(gh,kf) \right )\right ] \\ &=c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f)\overline{c(g,h,kf)}\overline{c_{R}(gh,k,f)}\\ &\quad \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h)u_R(gh, k)u_{R}(ghk,f)\right ). \end{split} \end{align} $$

Hence, we obtain

(2.59)$$ \begin{align} c_R(g,h,k) c_R(g,hk,f) c_R(h,k,f)\overline{c(g,h,kf)}\overline{c_{R}(gh,k,f)}=1,\quad\text{for all } g,h,k,f\in G. \end{align} $$

This means $c_{R}\in Z^{3}(G,{\mathbb T})$.

2.4 The $H^{3}(G,{\mathbb T})$-valued index

From the previous subsection, we remark the following fact:

Lemma 2.6. For any $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $, there are

(2.60)$$ \begin{align} \begin{split} \alpha\in \mathop{\mathrm{EAut}}\nolimits(\omega), \quad 0<\theta<\frac\pi 2,\quad (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits\left (\omega,\theta\right ),\quad (\eta_{g}^\sigma )\in {\mathcal T}\left(\theta, (\tilde\beta_g )\right),\quad (\alpha_L,\alpha_R,\Theta)\in{\mathfrak D}^{\theta}_\alpha,\\ ( (W_g), (u_R(g,h)))\in\mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ). \end{split} \end{align} $$

Proof. Because $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $, there is some $0<\theta <\frac \pi 2$ such that $\mathop {\mathrm {IG}}\nolimits (\omega ,\theta )\neq \emptyset $, and hence $\left (\tilde \beta _g\right )\in \mathop {\mathrm {IG}}\nolimits (\omega ,\theta )$ and $(\eta _g^\sigma )\in {\mathcal T}\left (\theta , \left (\tilde \beta _g\right )\right )$ exist. Because $\omega \in \mathop {\mathcal {SL}}\nolimits $, by definition there exists some $\alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$, and by the definition of $\mathop {\mathrm {EAut}}\nolimits (\omega )$, there is some $(\alpha _L,\alpha _R,\Theta )\in {\mathfrak D}^{\theta }_\alpha $. The existence of $( (W_g), (u_R(g,h)))\in \mathop {\mathrm {IP}}\nolimits ( \omega , \alpha , \theta , (\tilde \beta _g ), (\eta _{g}^\sigma ), (\alpha _L,\alpha _R,\Theta ) )$ is given by Lemma 2.1.

By Lemma 2.4, for $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $ and each choice of (2.60), we can associate some element of $H^3(G,{\mathbb T})$:

(2.61)$$ \begin{align} h^{(1)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ). \end{align} $$

In this subsection, we show that the third cohomology class does not depend on the choice of (2.60):

Theorem 2.7. For any $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $,

$$\begin{align*}h^{(1)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ) \end{align*}$$

is independent of the choice of

$$\begin{align*}\alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta), ( (W_g), (u_\sigma(g,h))). \end{align*}$$

Definition 2.8. Let $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $. We denote the third cohomology given in Theorem 2.7 by

$$\begin{align*}h(\omega):=h^{(1)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ). \end{align*}$$

First we show the independence from $( (W_g), (u_\sigma (g,h)))$.

Lemma 2.9. Set

(2.62)$$ \begin{align} &\omega\in\mathop{\mathcal{SL}}\nolimits, \enspace \alpha\in \mathop{\mathrm{EAut}}\nolimits(\omega), \enspace 0<\theta<\frac\pi 2,\quad (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta), \quad (\eta_{g}^\sigma )\in {\mathcal T}\left (\theta, (\tilde\beta_g )\right ),\quad (\alpha_L,\alpha_R,\Theta)\in {\mathcal D}_{\alpha}^{\theta}, \end{align} $$

(2.63)$$ \begin{align} &( (W_g), (u_\sigma(g,h))), ( (\tilde W_g ), (\tilde u_\sigma(g,h)))\in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ). \end{align} $$

Then we have

(2.64)$$ \begin{align} &h^{(1)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right )\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad= h^{(1)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (\tilde W_g ), (\tilde u_\sigma(g,h))) \right ). \end{align} $$

Definition 2.10. From this lemma and because there is always $( (W_g), (u_R(g,h)))$ in $\mathop {\mathrm {IP}}\nolimits ( \omega , \alpha , \theta , (\tilde \beta _g ), (\eta _{g}^\sigma ), (\alpha _L,\alpha _R,\Theta ) )$ by Lemma 2.1, we may define

(2.65)$$ \begin{align} h^{(2)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ) := h^{(1)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ) \end{align} $$

for any

(2.66)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \enspace \alpha\in \mathop{\mathrm{EAut}}\nolimits(\omega), \enspace 0<\theta<\frac\pi 2,\enspace (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta), \enspace (\eta_{g}^\sigma )\in {\mathcal T}\left (\theta, (\tilde\beta_g )\right ),\enspace (\alpha_L,\alpha_R,\Theta)\in {\mathcal D}_{\alpha}^{\theta}, \end{align} $$

independent of the choice of $( (W_g), (u_\sigma (g,h)))$.

Proof. Because

(2.67)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g)\circ\pi_0 &=\pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1} =\mathop{\mathrm{Ad}}\nolimits\left ( \tilde W_g\right )\circ\pi_0, \end{align} $$

(2.68)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_R(g,h)\right )\circ\pi_R &=\pi_R\circ\alpha_R\circ\eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1} \circ\alpha_R^{-1}=\mathop{\mathrm{Ad}}\nolimits\left ( \tilde u_R(g,h)\right )\circ\pi_R \end{align} $$

and $\pi _{0}, \pi _{R}$ are irreducible, there are $b(g),a(g,h)\in {\mathbb T}$, $g,h\in G$, such that

(2.69)$$ \begin{align} W_g=b(g)\tilde W_g,\qquad \tilde u_R(g,h)=a(g,h)u_R(g,h). \end{align} $$

Set

(2.70)$$ \begin{align} c_{R}&:=c_R\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ),\notag\\ \tilde c_R&:=c_{R}\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (\tilde W_g ), (\tilde u_\sigma(g,h))) \right ). \end{align} $$

Then from the definition of these values and equation (2.69), we have

(2.71)$$ \begin{align} a(g,h)a(gh,k)&\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h) u_R(gh,k)\right ) = \mathbb I_{{\mathcal H}_L}\otimes \tilde u_R(g,h) \tilde u_R(gh,k)\notag\\ &=\tilde c_R(g,h,k) \left ( \tilde W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes \tilde u_R(h,k)\right ) \tilde W_g^*\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes \tilde u_R(g,hk)\right )\notag\\ &=\tilde c_R(g,h,k)a(h,k)a(g,hk) \left ( W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_{g}^{*}\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)\right )\notag\\ &=\tilde c_R(g,h,k)a(h,k)a(g,hk)\overline{c_{R}(g,h,k)} \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,h) u_R(gh,k)\right ). \end{align} $$

Hence we have $\tilde c_{R}(g,h,k)=c_R(g,h,k)\overline {a(h,k)a(g,hk)}a(g,h)a(gh,k)$, and we get $[c_{R}]_{H^{3}\left (G,{\mathbb T}\right )}=\left [\tilde c_{R}\right ]_{H^{3}\left (G,{\mathbb T}\right )}$, proving the claim.

Next we show the independence from $\alpha , (\alpha _L,\alpha _R,\Theta )$:

Lemma 2.11. Set

(2.72)$$ \begin{align} &\omega\in\mathop{\mathcal{SL}}\nolimits, \quad \alpha_{1},\alpha_{2}\in \mathop{\mathrm{EAut}}\nolimits(\omega), \quad 0<\theta<\frac\pi 2,\ (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta), \quad (\eta_{g}^\sigma )\in {\mathcal T}\left (\theta, (\tilde\beta_g )\right ), \end{align} $$

(2.73)$$ \begin{align} & \left(\alpha_{L,1},\alpha_{R,1},\Theta_{1}\right)\in {\mathcal D}_{\alpha_{1}}^{\theta},\quad \left(\alpha_{L,2},\alpha_{R,2},\Theta_{2}\right)\in {\mathcal D}_{\alpha_{2}}^{\theta}. \end{align} $$

Then we have

(2.74)$$ \begin{align} h^{(2)} \left ( \omega, \alpha_{1} , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), \left(\alpha_{L,1},\alpha_{R,1},\Theta_{1}\right) \right ) = h^{(2)} \left ( \omega, \alpha_{2} , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), \left(\alpha_{L,2},\alpha_{R,2},\Theta_{2}\right) \right ). \end{align} $$

Definition 2.12. From this lemma and because there are always $ \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$ and $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta }$ for $\omega \in \mathop {\mathcal {SL}}\nolimits $ and $0<\theta <\frac \pi 2$ by the definition, we may define

(2.75)$$ \begin{align} h^{(3)} \left ( \omega, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ) \right ) := h^{(2)} \left ( \omega, \alpha , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_{L},\alpha_{R},\Theta) \right ) \end{align} $$

for any

(2.76)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \quad 0<\theta<\frac\pi 2,\quad (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta), \quad (\eta_{g}^\sigma )\in {\mathcal T}\left (\theta, (\tilde\beta_g )\right ), \end{align} $$

independent of the choice of $\alpha , (\alpha _L,\alpha _R,\Theta )$.

Proof. By Lemma 2.1, there are

(2.77)$$ \begin{align} \left ( \left(W_{g,1}\right), \left(u_{\sigma,1}(g,h)\right)\right ) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha_{1}, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), \left(\alpha_{L,1},\alpha_{R,1},\Theta_1\right) \right ). \end{align} $$

For each $i=1,2$, we have $\Theta _{i}\in \mathop {\mathrm {Aut}}\nolimits {\mathcal A}_{C_{\theta }^{c}}$ and

(2.78)$$ \begin{align} \alpha_{i}=({\textrm{{inner}}})\circ\alpha_{0,i}\circ\Theta_{i}, \end{align} $$

setting

(2.79)$$ \begin{align} \alpha_{0,i}:=\alpha_{L,i}\otimes \alpha_{R,i}. \end{align} $$

Because $\omega _{0}\circ \alpha _{1}=\omega =\omega _{0}\circ \alpha _{2}$, we have $\omega _{0}\circ \alpha _{2}\circ \alpha _{1}^{-1}=\omega _{0}$. Therefore, there is a unitary $\tilde V$ on ${\mathcal H}_{0}$ such that $ \pi _{0}\circ \alpha _{2}\circ \alpha _{1}^{-1}=\mathop {\mathrm {Ad}}\nolimits \left ( \tilde V\right )\circ \pi _{0} $. Substituting equation (2.78) into this, we see that there is a unitary V on ${\mathcal H}_{0}$ satisfying

(2.80)$$ \begin{align} \pi_{0}\circ \alpha_{0, 2}\circ\Theta_{2}=\mathop{\mathrm{Ad}}\nolimits\left ( V\right )\circ\pi_{0}\circ \alpha_{0, 1}\circ\Theta_{1}. \end{align} $$

From this, we obtain

(2.81)$$ \begin{align} \begin{split} &\mathop{\mathrm{Ad}}\nolimits\left ( V W_{g,1} V^{*}\right )\circ\pi_{0} \\ &\quad =\mathop{\mathrm{Ad}}\nolimits\left ( V W_{g,1}\right ) \pi_{0}\circ \alpha_{0, 1}\circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}\\ &\quad =\mathop{\mathrm{Ad}}\nolimits\left ( V\right )\circ \pi_{0} \circ\alpha_{0,1}\circ\Theta_{1}\circ\eta_g\beta_g^U\circ\Theta_{1}^{-1}\circ\alpha_{0,1}^{-1} \circ \alpha_{0, 1}\circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}\\ &\quad =\pi_{0}\circ\alpha_{0, 2}\circ\Theta_{2}\circ \Theta_{1}^{-1}\circ \alpha_{0, 1}^{-1} \circ\alpha_{0,1}\circ\Theta_{1}\circ\eta_g\beta_g^U\circ\Theta_{1}^{-1}\circ\alpha_{0,1}^{-1} \circ \alpha_{0, 1}\circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}\\ &\quad =\pi_{0}\circ\alpha_{0, 2}\circ\Theta_{2} \circ\eta_g\beta_g^U \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}, \end{split} \end{align} $$

for all $g\in G$. Furthermore, we have

(2.82)$$ \begin{align} &\mathop{\mathrm{Ad}}\nolimits\left ( V\left (\mathbb I_{{\mathcal H}_{L}}\otimes u_{R,1}(g,h)\right ) V^{*} \right )\circ\pi_{0} = \mathop{\mathrm{Ad}}\nolimits\left ( V\left (\mathbb I_{{\mathcal H}_{L}}\otimes u_{R,1}(g,h)\right ) \right )\circ \pi_{0}\circ \alpha_{0, 1}\circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}\notag\\ &=\mathop{\mathrm{Ad}}\nolimits\left ( V\right )\circ \pi_{0}\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_{L}}}\otimes \alpha_{R,1} \eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\alpha_{R,1}^{-1}\right ) \alpha_{0, 1}\circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}\notag\\ &= \pi_{0}\circ \alpha_{0, 2}\circ\Theta_{2}\circ \Theta_{1}^{-1} \circ \alpha_{0, 1}^{-1}\left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_{L}}}\otimes \alpha_{R,1} \eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\alpha_{R,1}^{-1}\right ) \circ\alpha_{0, 1}\circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1} \notag\\ &= \pi_{0}\circ \alpha_{0, 2}\circ\Theta_{2}\circ \Theta_{1}^{-1} \circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_{L}}}\otimes \eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\right ) \circ\Theta_{1} \circ \Theta_{2}^{-1}\circ \alpha_{0, 2}^{-1}. \end{align} $$

Now, because $\eta _g^R\beta _g^{R U} \eta _h^R\left (\beta _g^{R U}\right )^{-1}\left ( \eta _{gh}^R\right )^{-1}$ is an automorphism on ${\mathcal A}_{C_{\theta }}$ and $\Theta _{2}\circ \Theta _{1}^{-1}$ is an automorphism on ${\mathcal A}_{C_{\theta }^{c}}$, they commute. Therefore, we have

(2.83)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits&\left ( V\left (\mathbb I_{{\mathcal H}_{L}}\otimes u_{R,1}(g,h)\right ) V^{*} \right )\circ\pi_{0}\notag\\ &= \text{equation }(2.83)= \pi_{0}\circ \alpha_{0, 2} \circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_{L}}}\otimes \eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\right ) \circ \alpha_{0, 2}^{-1}\notag\\ &= \pi_{L}\otimes \left ( \pi_{R}\circ \alpha_{R,2}\eta_{g}^{R}\beta_{g}^{RU} \eta_{h}^{R}\left ( \beta_{g}^{RU}\right )^{-1} \left ( \eta_{gh}^{R}\right )^{-1}\left ( \alpha_{R,2}\right )^{-1}\right ). \end{align} $$

From this equality and the fact that $\pi _{L}$ is irreducible, we see that $V\left (\mathbb I_{{\mathcal H}_{L}}\otimes u_{R,1}(g,h)\right ) V^{*}$ is of the form $\mathbb I_{{\mathcal H}_{L}}\otimes u_{R,2}(g,h)$ with some unitary $u_{R,2}(g,h)$ on ${\mathcal H}_{R}$. This $u_{R,2}(g,h)$ satisfies

(2.84)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_{R,2}(g,h)\right )\circ\pi_{R} =\pi_{R}\circ \alpha_{R,2}\eta_{g}^{R}\beta_{g}^{RU} \eta_{h}^{R}\left ( \beta_{g}^{RU}\right )^{-1} \left ( \eta_{gh}^{R}\right )^{-1}\left ( \alpha_{R,2}\right )^{-1}. \end{align} $$

Analogously, we obtain a unitary $u_{L,2}(g,h)$ on ${\mathcal H}_{L}$ such that

(2.85)$$ \begin{align} V\left ( u_{L,1}(g,h)\otimes \mathbb I_{{\mathcal H}_{R}}\right ) V^{*}&= u_{L,2}(g,h)\otimes\mathbb I_{{\mathcal H}_{R}}, \end{align} $$

(2.86)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( u_{L,2}(g,h)\right )\circ\pi_{L} &=\pi_{L}\circ \alpha_{L,2}\eta_{g}^{L}\beta_{g}^{LU} \eta_{h}^{L}\left ( \beta_{g}^{LU}\right )^{-1} \left ( \eta_{gh}^{L}\right )^{-1}\left ( \alpha_{L,2}\right )^{-1}. \end{align} $$

From equations (2.81), (2.84) and (2.85), we see that

(2.87)$$ \begin{align} \left ( \left ( V W_{g,1} V^{*}\right ) , \left ( u_{\sigma,2}(g,h)\right )\right )\in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha_{2}, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), \left(\alpha_{L,2},\alpha_{R,2},\Theta\right) \right ). \end{align} $$

Set

(2.88)$$ \begin{align} c_{R,1}&:=c_R\left ( \omega, \alpha_{1}, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), \left(\alpha_{L,1},\alpha_{R,1},\Theta_{1}\right),\left ( \left(W_{g,1}\right), \left(u_{\sigma,1}(g,h)\right)\right ) \right ),\notag\\ c_{R,2}&:=c_{R}\left ( \omega, \alpha_{2}, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), \left(\alpha_{L,2},\alpha_{R,2},\Theta_{2}\right), \left ( V W_{g,1} V^{*}\right ) , \left ( u_{\sigma,2}(g,h)\right ) \right ). \end{align} $$

It suffices to show that $c_{R,1}=c_{R,2}$. This can be checked directly as follows:

(2.89)$$ \begin{align} V& \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R,1}(g,h) u_{R,1}(gh,k) \right ) V^{*} = \mathbb I_{{\mathcal H}_L}\otimes u_{R,2}(g,h) u_{R,2}(gh,k)\notag\\ &=c_{R,2}(g,h,k) \left ( V W_{g,1}V^{*}\left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R,2}(h,k)\right ) VW_{g,1}^*V^{*}\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R,2}(g,hk)\right )\notag\\ &=c_{R,2}(g,h,k)V \left ( W_{g,1}\left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R,1}(h,k)\right ) W_{g,1}^{*}\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R,1}(g,hk)\right ) V^{*}\notag\\ &= c_{R,2}(g,h,k)\overline{c_{R,1}(g,h,k)}V \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R,1}(g,h) u_{R,1}(gh,k) \right ) V^{*}. \end{align} $$

Lemma 2.13. Set

(2.90)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \quad 0<\theta<\frac\pi 2,\quad (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta), \quad (\eta_{g}^\sigma ), \left(\tilde \eta_{g}^\sigma\right)\in {\mathcal T}\left (\theta, (\tilde\beta_g )\right ). \end{align} $$

Then we have

(2.91)$$ \begin{align} h^{(3)} \left ( \omega , \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ) \right ) = h^{(3)} \left ( \omega, \theta, (\tilde\beta_g ), \left(\tilde \eta_{g}^\sigma\right) \right ). \end{align} $$

Definition 2.14. From this lemma and the definition of $ \mathop {\mathrm {IG}}\nolimits (\omega ,\theta )$, we may define

(2.92)$$ \begin{align} h^{(4)} \left ( \omega, \theta, (\tilde\beta_g ) \right ) := h^{(3)} \left ( \omega, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ) \right ) \end{align} $$

for any

(2.93)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \quad 0<\theta<\frac\pi 2,\quad (\tilde\beta_g )\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta), \quad (\eta_{g}^\sigma )\in {\mathcal T}\left (\theta, (\tilde\beta_g )\right ), \end{align} $$

independent of the choice of $(\eta _{g}^\sigma )$.

Proof. There are $ \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$ and $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta }$ for $\omega \in \mathop {\mathcal {SL}}\nolimits $ by the definition. We set $\alpha _0:=\alpha _L\otimes \alpha _R$ and $\eta _g:=\eta _g^L\otimes \eta _g^R$, $\tilde \eta _g:=\tilde \eta _g^L\otimes \tilde \eta _g^R$. By Lemma 2.1, there is some

(2.94)$$ \begin{align} \left ( \left(W_{g}\right), (u_{\sigma}(g,h))\right ) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_{L},\alpha_{R},\Theta) \right ). \end{align} $$

Because $ (\eta _{g}^\sigma ), \left (\tilde \eta _{g}^\sigma \right )\in {\mathcal T}\left (\theta , \left (\tilde \beta _g\right )\right )$, we have

(2.95)$$ \begin{align} \tilde\beta_g=({\textrm{{inner}}})\circ\left ( \eta_g^L\otimes\eta_g^R\right )\circ\beta_g^U =({\textrm{{inner}}}) \circ\left (\tilde \eta_g^L\otimes\tilde \eta_g^R\right )\circ\beta_g^U. \end{align} $$

From this, we obtain

(2.96)$$ \begin{align} \tilde \eta_g^L\circ\left (\eta_g^L\right )^{-1}\otimes\tilde \eta_g^R\circ\left (\eta_g^R\right )^{-1} =({\textrm{{inner}}}), \end{align} $$

hence there are unitaries $v_g^\sigma \in {\mathcal A}_{H_\sigma }$, $\sigma =L,R$, such that

(2.97)$$ \begin{align} \tilde \eta_g^\sigma\circ\left (\eta_g^\sigma\right )^{-1} =\mathop{\mathrm{Ad}}\nolimits\left ( v_g^\sigma\right ). \end{align} $$

Because $\tilde \eta _g^\sigma , \eta _g^\sigma $ are automorphisms on ${\mathcal A}_{C_\theta ,\sigma }$, $v_g^\sigma $ belongs to ${\mathcal A}_{C_\theta ,\sigma }$. (See Lemma B.1.) Setting $v_g:=v_g^L\otimes v_g^R$, we obtain $\tilde \eta _g=\mathop {\mathrm {Ad}}\nolimits \left ( v_g\right )\circ \eta _g$.

Set

(2.98)$$ \begin{align} \tilde W_g&:=\left ( \left ( \pi_L\alpha_L\left ( v_g^L\right )\right ) \otimes \left ( \pi_R\alpha_R\left ( v_g^R\right )\right )\right ) W_g, \end{align} $$

(2.99)$$ \begin{align} \tilde u_\sigma(g,h) &:= \pi_\sigma\left ( \alpha_\sigma\left ( v_g^\sigma \cdot \left ( \eta_g^\sigma\beta_g^{\sigma U}\right )\left ( v_h^\sigma\right )\right ) \right )\cdot u_\sigma\left ( g,h\right )\cdot \pi_\sigma\left (\alpha_\sigma\left ( \left ( v_{gh}^{\sigma}\right )^*\right )\right ), \end{align} $$

for each $g,h\in G$ and $\sigma =L,R$. We claim that

(2.100)$$ \begin{align} \left (\left ( \tilde W_g\right ), \left ( \tilde u_\sigma(g,h)\right ) \right ) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, (\tilde\beta_g ), \left(\tilde \eta_{g}^\sigma\right), (\alpha_L,\alpha_R,\Theta) \right ). \end{align} $$

First, we have

(2.101)$$ \begin{align} \begin{split} \pi_0&\circ\alpha_0\circ\Theta\circ\tilde \eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1} =\pi_0\circ\alpha_0\circ\Theta\circ \mathop{\mathrm{Ad}}\nolimits\left ( v_g\right )\circ\eta_g \beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1}\\ &=\pi_0\circ\alpha_0\circ\mathop{\mathrm{Ad}}\nolimits\left ( v_g\right )\circ\Theta \circ\eta_g \beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1}\\ &= \mathop{\mathrm{Ad}}\nolimits\left ( \left ( \pi_L\alpha_L\left ( v_g^L\right )\right ) \otimes \left ( \pi_R\alpha_R\left ( v_g^R\right )\right ) \right ) \pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1} = \mathop{\mathrm{Ad}}\nolimits\left ( \tilde W_g\right )\circ\pi_0. \end{split} \end{align} $$

For the first equality, we substituted $\tilde \eta _g=\mathop {\mathrm {Ad}}\nolimits \left ( v_g\right )\circ \eta _g$, and for the second equality, we used the fact that $v_g^\sigma $ belongs to ${\mathcal A}_{C_\theta ,\sigma }$, while $\Theta $ is an automorphism on ${\mathcal A}_{\left ( C_\theta \right )^c,\sigma }$. The last equality follows from the definition of $W_g$. On the other hand, we have

(2.102)$$ \begin{align} &\pi_{\sigma}\circ\alpha_{\sigma}\circ{\tilde\eta}_g^{\sigma}\beta_g^{{\sigma} U} {\tilde\eta}_h^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}\left ( {\tilde\eta}_{gh}^{\sigma}\right )^{-1} \circ\alpha_{\sigma}^{-1}\notag\\ &= \pi_{\sigma}\circ\alpha_{\sigma}\circ\mathop{\mathrm{Ad}}\nolimits\left ( v_g^\sigma\right )\circ\eta_g^{\sigma}\beta_g^{{\sigma} U} \mathop{\mathrm{Ad}}\nolimits\left ( v_h^\sigma\right )\circ\eta_h^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}( \eta_{gh}^{\sigma})^{-1} \mathop{\mathrm{Ad}}\nolimits\left ( {v_{gh}^\sigma}^*\right ) \circ\alpha_{\sigma}^{-1}\notag\\ &= \mathop{\mathrm{Ad}}\nolimits\left ( \pi_{\sigma}\circ\alpha_{\sigma} \left ( \left ( v_g^\sigma\right ) \eta_g^{\sigma}\beta_g^{{\sigma} U}\left ( v_h^\sigma\right )\right ) \right ) \pi_{\sigma}\circ\alpha_{\sigma}\eta_g^{\sigma}\beta_g^{{\sigma} U} \eta_h^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}( \eta_{gh}^{\sigma})^{-1} \circ\alpha_{\sigma}^{-1}\circ \mathop{\mathrm{Ad}}\nolimits\left (\alpha_\sigma\left ( {v_{gh}^\sigma}^*\right )\right )\notag\\ &= \mathop{\mathrm{Ad}}\nolimits\left ( \pi_{\sigma}\circ\alpha_{\sigma} \left ( \left ( v_g^\sigma\right ) \eta_g^{\sigma}\beta_g^{{\sigma} U}\left ( v_h^\sigma\right )\right )\right ) \circ \mathop{\mathrm{Ad}}\nolimits\left ( u_\sigma(g,h)\right ) \pi_\sigma\circ \mathop{\mathrm{Ad}}\nolimits\left (\alpha_\sigma\left ( {v_{gh}^\sigma}^*\right )\right ) = \mathop{\mathrm{Ad}}\nolimits\left ( \tilde u_{\sigma}(g,h)\right )\circ\pi_{\sigma}, \end{align} $$

for all $g,h\in G$. For the first equality, we substituted $\tilde \eta _g=\mathop {\mathrm {Ad}}\nolimits \left ( v_g\right )\circ \eta _g$. The third equality is the definition of $u(g,h)$. Hence we have proven formula (2.100).

Set

(2.103)$$ \begin{align} c_{R}&:=c_R\left ( \omega, \alpha, \theta, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta),( (W_g), (u_\sigma(g,h))) \right ),\notag\\ \tilde c_R&:=c_{R}\left ( \omega, \alpha, \theta, (\tilde\beta_g ), \left(\tilde \eta_{g}^\sigma\right), (\alpha_L,\alpha_R,\Theta),( (\tilde W_g ), (\tilde u_\sigma(g,h))) \right ). \end{align} $$

In order to show the statement of the lemma, it suffices to show that $c_R=\tilde c_R$. Substituting the definition of $\tilde u_R$, we obtain

(2.104)$$ \begin{align} \begin{split} &\tilde u_R(g,h) \tilde u_R(gh,k)\\ &\quad = \pi_R\left ( \alpha_R\left ( v_g^R \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\right ) \right )\cdot u_R\left ( g,h\right )\cdot \pi_R\left (\alpha_R\left ( \left ( v_{gh}^{R}\right )^*\right )\right )\\ &\pi_R\left ( \alpha_R\left ( v_{gh}^R \left ( \eta_{gh}^R\beta_{gh}^{R U}\right )( v_k^R)\right ) \right )\cdot u_R\left ( gh,k\right )\cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right ) \\ &\quad = \pi_R\left ( \alpha_R\left ( v_g^R \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\right ) \right ) \\ &\qquad \cdot \left [ u_R\left ( g,h\right )\cdot \pi_R\left ( \alpha_R\left ( \left ( \eta_{gh}^R\beta_{gh}^{R U}\right )( v_k^R)\right ) \right ) \right ] u_R\left ( gh,k\right ) \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right ) \\ &\quad = \pi_R\left ( \alpha_R\left ( v_g^R \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\right ) \right )\\ &\quad \cdot \left [ \mathop{\mathrm{Ad}}\nolimits\left ( u_R\left ( g,h\right )\right )\left ( \pi_R\left ( \alpha_R\left ( \left ( \eta_{gh}^R\beta_{gh}^{R U}\right )( v_k^R)\right ) \right )\right ) \cdot u_R\left ( g,h\right ) \right ] u_R\left ( gh,k\right )\cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right ) \\ &\quad = \pi_R\left ( \alpha_R\left ( v_g^R \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\right ) \right ) \\ &\qquad \cdot \left ( \pi_R\left (\alpha_R\circ\eta_g^R\beta_g^{R U} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1} \circ\alpha_R^{-1} \alpha_R\left ( \left ( \eta_{gh}^R\beta_{gh}^{R U}\right )( v_k^R)\right ) \right )\right )\\ &\qquad \cdot u_R\left ( g,h\right ) u_R\left ( gh,k\right )\cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right )\\ &\quad =\pi_R\left ( \alpha_R\left ( v_g^R \cdot \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\cdot \eta_g^R\beta_g^{R U} \eta_h^R\beta_{h}^{R U}( v_k^R)\right ) \right )\cdot u_R\left ( g,h\right ) u_R\left ( gh,k\right )\cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right ). \end{split}\end{align} $$

For the fourth equality, we used the definition of $u_R$. From this equation, applying equation (2.52) to the $[\cdot ]$ part, we have

(2.105)$$ \begin{align} &\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R(g,h) \tilde u_R(gh,k)\notag\\ &\quad = \mathbb I_{{\mathcal H}_L}\otimes \pi_R\left ( \alpha_R\left ( v_g^R \cdot \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\cdot \eta_g^R\beta_g^{R U} \eta_h^R\beta_{h}^{R U}( v_k^R)\right ) \right ) \notag\\ &\qquad \cdot \left [ u_R\left ( g,h\right ) u_R\left ( gh,k\right )\right ]\cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right )\notag\\ &\quad =c_R(g,h,k)\left ( \mathbb I_{{\mathcal H}_L}\otimes \pi_R\left ( \alpha_R\left ( v_g^R \cdot \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\cdot \eta_g^R\beta_g^{R U} \eta_h^R\beta_{h}^{R U}( v_k^R)\right ) \right )\right )\notag\\ &\qquad \left\{ W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\right\} \left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)\right ) \cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right ). \end{align} $$

Now from the definition of $\tilde u_R$, the $\{\cdot \}$ part becomes

(2.106)$$ \begin{align} W_g&\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^* \nonumber\\ &= \mathop{\mathrm{Ad}}\nolimits( W_g)\circ\pi_0\circ\left ( \mathop{\mathrm{id}}\nolimits_L\otimes \alpha_R\left ( \left ( v_h^R \cdot \eta_h^R\beta_h^{R U}( v_k^R)\right )^*\right ) \right ) \nonumber \\& \quad \cdot \mathop{\mathrm{Ad}}\nolimits( W_g)\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\cdot\left ( \mathop{\mathrm{Ad}}\nolimits( W_g)\pi_0\left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\left ( v_{hk}^{R}\right )\right ) \right ). \end{align} $$

Because $v_g^R$ belongs to ${\mathcal A}_{C_\theta ,R}$ and $\eta _g^R$ is an automorphism on ${\mathcal A}_{C_\theta ,R}$ while $\Theta $ is an automorphism on ${\mathcal A}_{\left ( C_\theta \right )^c}$ and $\beta _g^U\left ( {\mathcal A}_{C_\theta ,R}\right )={\mathcal A}_{C_\theta ,R}$, we have

(2.107)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g)&\circ\pi_0\circ\left ( \mathop{\mathrm{id}}\nolimits_L\otimes \alpha_R\left ( \left ( v_h^R \cdot \eta_h^R\beta_h^{R U}( v_k^R)\right )^*\right ) \right )\notag\\ &= \pi_0\circ\alpha_0\circ\Theta\circ\eta_g\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1} \circ \left ( \mathop{\mathrm{id}}\nolimits_L\otimes \alpha_R\left ( \left ( v_h^R \cdot \eta_h^R\beta_h^{R U}( v_k^R)\right )^*\right ) \right )\notag\\ &= \pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( \left ( v_h^R \cdot \eta_h^R\beta_h^{R U}( v_k^R)\right )^*\right ) \right ),\quad \text{and}\notag\\ \mathop{\mathrm{Ad}}\nolimits( W_g)&\circ\pi_0\left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\left ( v_{hk}^{R}\right )\right ) =\pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( v_{hk}^{R} \right ) \right ). \end{align} $$

Substituting this into equation (2.106), we obtain

(2.108)$$ \begin{align} &W_g\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(h,k)\right ) W_g^*\nonumber\\ &\quad = \pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( \left ( v_h^R \cdot \eta_h^R\beta_h^{R U}( v_k^R)\right )^*\right ) \right ) \notag\\ &\qquad\cdot \mathop{\mathrm{Ad}}\nolimits( W_g)\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\cdot \pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( v_{hk}^{R} \right ) \right ). \end{align} $$

Substituting this to the $\{ \}$ part and the $\{\cdot \}$ part of equation (2.105), we obtain

(2.109)$$ \begin{align} \mathbb I_{{\mathcal H}_L}&\otimes \tilde u_R(g,h) \tilde u_R(gh,k)\notag\\ &=c_R(g,h,k)\left ( \mathbb I_{{\mathcal H}_L}\otimes \pi_R\left ( \alpha_R\left ( v_g^R \cdot \left ( \eta_g^R\beta_g^{R U}\right )( v_h^R)\cdot \eta_g^R\beta_g^{R U} \eta_h^R\beta_{h}^{R U}( v_k^R)\right ) \right )\right )\notag\\ & \quad \pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( \left ( v_h^R \cdot \eta_h^R\beta_h^{R U}( v_k^R)\right )^*\right ) \right ) \nonumber \\ &\quad \cdot \mathop{\mathrm{Ad}}\nolimits( W_g)\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\cdot \pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( v_{hk}^{R} \right ) \right )\notag\\ & \quad\left ( \mathbb I_{{\mathcal H}_L}\otimes u_R(g,hk)\cdot \pi_R\left (\alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right )\right )\right ) \notag\\ & =c_R(g,h,k)\left ( \mathbb I_{{\mathcal H}_L}\otimes \pi_R\left ( \alpha_R\left ( v_g^R \right ) \right )\right ) \mathop{\mathrm{Ad}}\nolimits( W_g)\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\cdot \pi_0 \left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\circ \eta_g^R\beta_g^{R U} \left ( v_{hk}^{R} \right ) \right )\notag\\ & \quad \pi_0\circ\left ( \mathop{\mathrm{id}}\nolimits_L\otimes \alpha_R\left ( \left ( v_g^R \cdot \eta_g^R\beta_g^{R U}\left ( v_{hk}^R\right )\right )^*\right ) \right ) \nonumber \\ &\qquad \cdot \left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R\left ( g,hk\right )\cdot\right ) \pi_0\left ( \mathop{\mathrm{id}}\nolimits_L\otimes\alpha_R\left ( v_{ghk}^{R}\right ) \alpha_R\left ( \left ( v_{ghk}^{R}\right )^*\right ) \right ) \notag\\ &= c_R(g,h,k) \mathop{\mathrm{Ad}}\nolimits\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes \pi_R\left ( \alpha_R\left ( v_g^R \right ) \right )\right ) W_g\right )\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\cdot \left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R\left ( g,hk\right )\right ) \notag\\ &=c_R(g,h,k) \left\{\mathop{\mathrm{Ad}}\nolimits\left ( \pi_L\left ( \alpha_L\left ({ v_g^L}^* \right )\otimes\mathbb I_{{\mathcal H}_R} \right )\right ) \mathop{\mathrm{Ad}}\nolimits \tilde W_g\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\right\}\cdot \left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R\left ( g,hk\right )\right ). \notag\\ \end{align} $$

Because of Lemma 2.3(iii), the $\{\cdot \}$ part of the last equation is equal to $\mathop {\mathrm {Ad}}\nolimits \tilde W_g\left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )$. Hence we obtain

(2.110)$$ \begin{align} \mathbb I_{{\mathcal H}_L}\otimes \tilde u_R(g,h) \tilde u_R(gh,k) =c_R(g,h,k) \mathop{\mathrm{Ad}}\nolimits \tilde W_g \left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R( h,k)\right )\cdot \left (\mathbb I_{{\mathcal H}_L}\otimes \tilde u_R\left ( g,hk\right )\right ). \end{align} $$

This proves $c_R=\tilde c_R$, completing the proof.

Lemma 2.15. Set

(2.111)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \quad 0<\theta<\frac\pi 2,\quad \left(\tilde\beta_{g}^{(1)}\right), \left(\tilde\beta_g^{(2)}\right)\in \mathop{\mathrm{IG}}\nolimits(\omega,\theta). \end{align} $$

Then we have

(2.112)$$ \begin{align} h^{(4)} \left ( \omega , \theta, \left(\tilde\beta_g^{(1)}\right) \right ) = h^{(4)} \left ( \omega, \theta, \left(\tilde\beta_g^{(2)}\right) \right ). \end{align} $$

Definition 2.16. From this lemma we may define

(2.113)$$ \begin{align} h^{(5)} \left ( \omega, \theta \right ) := h^{(4)} \left ( \omega, \theta, (\tilde\beta_g ) \right ) \end{align} $$

for any

(2.114)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \quad 0<\theta<\frac\pi 2,\quad \text{with } \mathop{\mathrm{IG}}\nolimits(\omega,\theta)\neq \emptyset, \end{align} $$

independent of the choice of $\left (\tilde \beta _g\right )$.

Proof. By the definition of $\mathop {\mathrm {IG}}\nolimits (\omega ,\theta )$, there are

(2.115)$$ \begin{align} \left(\eta_{g,i}^{\sigma}\right)_{g\in G, \, \sigma=L,R} \in {\mathcal T}\left(\theta, \left(\tilde\beta_g^{(i)}\right)\right),\quad \text{for } i=1,2. \end{align} $$

We set $\eta _{g,i}:=\eta _{g,i}^L\otimes \eta _{g,i}^R$, for $i=1,2$. There are $ \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$ and $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta }$ for $\omega \in \mathop {\mathcal {SL}}\nolimits $ by the definition. Setting $\alpha _0:=\alpha _L\otimes \alpha _R$, we have $\alpha =({\textrm {{inner}}})\circ \alpha _0\circ \Theta $. By Lemma 2.1, there is some

(2.116)$$ \begin{align} \left ( \left(W_{g,1}\right), \left(u_{\sigma}^{(1)}(g,h)\right)\right ) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, \left(\tilde\beta_g^{(1)}\right), \left(\eta_{g,1}^\sigma\right), (\alpha_{L},\alpha_{R},\Theta) \right ). \end{align} $$

Set

(2.117)$$ \begin{align} K^\sigma_g:=\eta_{g,2}^\sigma\circ \left ( \eta_{g,1}^\sigma\right )^{-1}\in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{C_\theta,\sigma}\right ),\quad\text{for } \sigma=L,R,\ g\in G,\qquad K_g:=K_{g}^L\otimes K_g^R\in\mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{C_\theta}\right ). \end{align} $$

We claim that there are unitaries $V_g^\sigma $, $g\in G, \ \sigma =L,R$, on ${\mathcal H}_\sigma $ such that

(2.118)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits \left ( V_g^\sigma\right )\circ\pi_\sigma =\pi_\sigma\circ\alpha_\sigma\circ K_g^\sigma\circ \left (\alpha_\sigma\right )^{-1}. \end{align} $$

To see this, note that

(2.119)$$ \begin{align} \omega=\omega\circ\tilde \beta_g^{(i)} =\omega_0\circ\alpha\circ\tilde \beta_g^{(i)} \sim_{\text{q.e.}} \omega_0\circ\alpha_0\circ\Theta\circ \left ( \eta_{g,i}^{L}\otimes \eta_{g,i}^{R} \right )\circ\beta_g^U,\quad i=1,2. \end{align} $$

Therefore, we have

(2.120)$$ \begin{align} \omega_0\circ\alpha_0\circ\Theta\circ \left ( \eta_{g,1}^{L}\otimes \eta_{g,1}^{R} \right ) \sim_{\text{q.e.}} \omega\circ\left ( \beta_g^U\right )^{-1} \sim_{\text{q.e.}} \omega_0\circ\alpha_0\circ\Theta\circ \left ( \eta_{g,2}^{L}\otimes \eta_{g,2}^{R} \right ), \end{align} $$

and then using the facts that $\Theta \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{C_\theta ^c}\right )$ and $K_g\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{C_\theta }\right )$,

(2.121)$$ \begin{align} \omega_0\sim_{\text{q.e.}} \omega_0\circ\alpha_0\circ\Theta\circ K_g\circ\Theta^{-1}\circ \alpha_0^{-1} =\omega_0\circ \alpha_0\circ K_g\circ \left ( \alpha_0\right )^{-1} =\bigotimes_{\sigma=L,R} \omega_\sigma \circ\alpha_\sigma K_g^\sigma \left ( \alpha_\sigma\right )^{-1}. \end{align} $$

This implies that $\omega _\sigma $ and $ \omega _\sigma \circ \alpha _\sigma K_g^\sigma \left ( \alpha _\sigma \right )^{-1}$ are quasiequivalent. Because $\pi _\sigma $ is irreducible, this implies the existence of a unitary $V_g^\sigma $ on ${\mathcal H}_\sigma $ satisfying equation (2.118), proving the claim.

Next we claim that there are unitaries $v^\sigma _{g,h}$ on ${\mathcal H}_\sigma $, for $g,h\in G$ and $\sigma =L,R$, such that

(2.122)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits_{W_{g,1}}\left (\mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) =\mathbb I_{{\mathcal H}_L}\otimes v^{R}_{g,h},\qquad \mathop{\mathrm{Ad}}\nolimits_{W_{g,1}}\left ( V_h^L\otimes \mathbb I_{{\mathcal H}_R}\right ) =v^{L}_{g,h}\otimes\mathbb I_{{\mathcal H}_R} \end{align} $$

and

(2.123)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( V_g^\sigma v^\sigma_{g,h} u_\sigma^{(1)}(g,h) \left ( V_{gh}^\sigma \right )^* \right )\pi_\sigma =\pi_\sigma\circ \alpha_\sigma \circ \eta_{g,2}^{\sigma}\beta_g^{{\sigma} U} \eta_{h,2}^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}\left ( \eta_{gh,2}^{\sigma}\right )^{-1}\circ \alpha_\sigma ^{-1}, \end{align} $$

for any $g,h\in G$ and $\sigma =L,R$. To see this, first we calculate

(2.124)$$ \begin{align}& \mathop{\mathrm{Ad}}\nolimits\left ( W_{g,1}\left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left ( W_{g,1}\right )^* \right ) \circ\pi_0 =\mathop{\mathrm{Ad}}\nolimits\left ( W_{g,1}\left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \right )\pi_0\circ \alpha_0\circ\Theta\circ\left ( {\eta_{g,1}}\beta_g^U\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}\notag\\ &\quad =\pi_0\circ \alpha_0\circ\Theta\circ {\eta_{g,1}}\beta_g^U\circ\Theta^{-1}\circ\alpha_0^{-1} \circ \left ( \mathop{\mathrm{id}}\nolimits_{L}\otimes \alpha_R\circ K_h^R\circ \left (\alpha_R\right )^{-1} \right ) \circ \alpha_0\circ\Theta\circ\left ( {\eta_{g,1}}\beta_g^U\right )^{-1}\!\circ\Theta^{-1}\circ\alpha_0^{-1}\notag\\ &\quad = \pi_0\circ \alpha_0\circ\Theta\circ {\eta_{g,1}}\beta_g^U\circ\Theta^{-1} \circ \left ( \mathop{\mathrm{id}}\nolimits_{L}\otimes K_h^R \right ) \circ\Theta\circ\left ( {\eta_{g,1}}\beta_g^U\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1} \notag\\ &\quad = \pi_0\circ \alpha_0\circ\Theta\circ {\eta_{g,1}}\beta_g^U \circ \left ( \mathop{\mathrm{id}}\nolimits_{L}\otimes K_h^R \right ) \circ\left ( {\eta_{g,1}}\beta_g^U\right )^{-1}\circ\Theta^{-1}\circ\alpha_0^{-1}\notag\\ &\quad =\pi_0\circ\alpha_0\circ\Theta \circ \left ( \mathop{\mathrm{id}}\nolimits_{L}\otimes {\eta_{g,1}^R}\beta_g^{RU} K_h^R \left ( {\eta_{g,1}^R}\beta_g^{RU}\right )^{-1} \right ) \circ\Theta^{-1}\circ\alpha_0^{-1}\notag\\ &\quad = \pi_0\circ \left ( \mathop{\mathrm{id}}\nolimits_{L}\otimes \alpha_R\circ {\eta_{g,1}^R}\beta_g^{RU} K_h^R \left ( {\eta_{g,1}^R}\beta_g^{RU}\right )^{-1}\alpha_R^{-1} \right ). \end{align} $$

In the fourth and sixth equalities, we used the fact that $K_h^R, \eta _{g,1}^R\beta _g^{RU} K_h^R \left ( \eta _{g,1}^R\beta _g^{RU}\right )^{-1}\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{C_\theta }\right )$ and $\Theta \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{C_\theta ^c}\right )$ commute, in order to remove $\Theta $. Equation (2.124) and the fact that $\pi _L$ is irreducible imply that there is a unitary $v^R_{g,h}$ satisfying equation (2.122). The same argument implies the existence of $v^L_{g,h}$ satisfying equation (2.122).

For this $v^R_{g,h}$, we would like to show equation (2.123). Rewriting

(2.125)$$ \begin{align} \begin{split} &\eta_{g,2}^{\sigma}\beta_g^{{\sigma} U} \eta_{h,2}^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}\left ( \eta_{gh,2}^{\sigma}\right )^{-1} \\ &\qquad\qquad\qquad=K_g^\sigma\circ\left ( \eta_{g,1}^{\sigma}\beta_g^{{\sigma} U} K_h^\sigma \left ( \eta_{g,1}^{\sigma}\beta_g^{{\sigma} U}\right )^{-1} \right )\circ \eta_{g,1}^{\sigma}\beta_g^{{\sigma} U} \eta_{h,1}^{\sigma}\left (\beta_g^{{\sigma} U}\right )^{-1}\left ( \eta_{gh,1}^{\sigma}\right )^{-1} \circ \left ( K_{gh}^{\sigma} \right )^{-1}, \end{split} \end{align} $$

we obtain

(2.126)$$ \begin{align} &\pi_L\otimes \pi_R\circ \alpha_R \eta_{g,2}^{R}\beta_g^{{R} U} \eta_{h,2}^{R}\left (\beta_g^{{R} U}\right )^{-1}\left ( \eta_{gh,2}^{R}\right )^{-1} \alpha_R ^{-1}\notag\\ &\quad =\pi_0\circ\left ( \mathop{\mathrm{id}}\nolimits_L\otimes \alpha_R\circ K_g^R\circ \left ( \eta_{g,1}^{R}\beta_g^{{R} U} K_h^R \left ( \eta_{g,1}^{R}\beta_g^{{R} U}\right )^{-1} \right ) \right.\nonumber \\ &\qquad\left.\, \circ\ \eta_{g,1}^{R}\beta_g^{{R} U} \eta_{h,1}^{R}\left (\beta_g^{{R} U}\right )^{-1}\left ( \eta_{gh,1}^{R}\right )^{-1} \circ \left ( K_{gh}^{R} \right )^{-1} \alpha_R^{-1} \right )\notag\\ &\quad = \pi_L\otimes \mathop{\mathrm{Ad}}\nolimits\left ( {V_g^R v_{g,h}^R u_R^{(1)}(g,h) \left ( V_{gh}^R\right )^*}\right ) \pi_R, \end{align} $$

substituting equations (2.118), (2.124) and (2.122). This proves equation (2.123) for $\sigma =R$. An analogous result for $\sigma =L$ can be proven by the same argument. Hence we have proven the claim (2.124) and (2.123).

Setting

(2.127)$$ \begin{align} V_g:=V_g^L\otimes V_g^R\in{\mathcal U}({\mathcal H}_0), \end{align} $$

we have

(2.128)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( V_gW_{g,1}\right )\circ\pi_0 &=\pi_0\circ \alpha_0\circ K_g\circ \alpha_0^{-1}\circ \alpha_0\circ\Theta\circ \eta_{g,1} \circ\beta_g^U\circ \Theta^{-1}\circ \alpha_0^{-1}\notag\\ &=\pi_0\circ \alpha_0\circ\Theta\circ \eta_{g,2}\circ\beta_g^U\circ\Theta^{-1}\circ \alpha_0^{-1}. \end{align} $$

In the last equality, we used the definition of $K_g$ and the commutativity of $\Theta $ and $K_g$ again. From equations (2.128) and (2.123), setting

(2.129)$$ \begin{align} u_\sigma^{(2)}(g,h):= V_g^\sigma v^\sigma_{g,h} u_\sigma^{(1)}(g,h) \left ( V_{gh}^\sigma \right )^*, \end{align} $$

we see that

(2.130)$$ \begin{align} \left ( \left ( V_gW_{g,1}\right ), \left ( {u_{R}^{(2)}}(g,h )\right )\right ) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta, \left(\tilde\beta_g^{(2)}\right), \left(\eta_{g,2}^\sigma\right), (\alpha_{L},\alpha_{R},\Theta) \right ) \end{align} $$

and

(2.131)$$ \begin{align} \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(2)}}(g,h ) =\left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left ( W_{g,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(1)}}(g,h )\left ( V_{gh}^R\right )^* \right ). \end{align} $$

Now we set

(2.132)$$ \begin{align} c_{R,1}&:=c_R\left ( \omega, \alpha, \theta, \left(\tilde\beta_g^{(1)}\right), \left(\eta_{g,1}^\sigma\right), (\alpha_{L},\alpha_{R},\Theta),\left ( (W_{g,1}), \left(u_{\sigma}^{(1)}(g,h)\right)\right ) \right ),\notag\\ c_{R,2}&:=c_{R}\left ( \omega, \alpha, \theta, \left(\tilde\beta_g^{(2)}\right), \left(\eta_{g,2}^\sigma\right), (\alpha_{L},\alpha_{R},\Theta), \left ( \left ( V_gW_{g,1}\right ), \left ( {u_{R}^{(2)}}(g,h )\right )\right ) \right ). \end{align} $$

To prove the Lemma, it suffices to show $c_{R,1}=c_{R,2}$. By equation (2.131), we have

(2.133)$$ \begin{align} \mathbb I_{{\mathcal H}_L}&\otimes u_R^{(2)}(g,h) u_R^{(2)}(gh,k) \nonumber\\ &=\left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left ( W_{g,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(1)}}(g,h )\left ( V_{gh}^R\right )^* \right )\nonumber\\& \quad \cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes V_{gh}^R\right ) W_{gh,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_k^R\right ) \left ( W_{gh,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(1)}}(gh,k )\left ( V_{ghk}^R\right )^* \right )\nonumber\\ &= \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left ( W_{g,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(1)}}(g,h ) \right )\nonumber\\& \quad \cdot W_{gh,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_k^R\right ) \left ( W_{gh,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(1)}}(gh,k )\left ( V_{ghk}^R\right )^* \right )\nonumber\\ &= \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left ( W_{g,1}\right )^* \left\{\mathop{\mathrm{Ad}}\nolimits\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes {u_{R}^{(1)}}(g,h ) \right )\cdot W_{gh,1} \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes V_k^R\right ) \right\}\nonumber\\ & \quad \cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes \left [ {u_{R}^{(1)}}(g,h ){u_{R}^{(1)}}(gh,k )\right ]\left ( V_{ghk}^R\right )^* \right )\nonumber\\ & =c_{R,1}(g,h,k) \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left ( W_{g,1}\right )^* \left\{\mathop{\mathrm{Ad}}\nolimits\left ( W_{g,1}W_{h,1} \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes V_k^R\right ) \right\}\nonumber\\ & \quad \cdot \left ( W_{g,1}\left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(1)}(h,k)\right ) W_{g,1}^*\right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(1)}(g,hk)\left ( V_{ghk}^R\right )^*\right ). \end{align} $$

We used equation (2.52) for the $[\cdot ]$ part and Lemma 2.3(ii) and equation (2.122) for the $\{\cdot \}$ part in the fourth equality. Again using equation (2.131), we have

(2.134)$$ \begin{align} & \mathbb I_{{\mathcal H}_L} \otimes u_R^{(2)}(g,h) u_R^{(2)}(gh,k)= \text{equation }(2.134)\hspace{220pt} \nonumber \\ & =c_{R,1}(g,h,k) \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right ) \left\{\mathop{\mathrm{Ad}}\nolimits\left ( W_{h,1} \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes V_k^R\right ) \right\}\notag\\ & \quad \cdot\left ( W_{h,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes \left ( V_k^R\right )^*\right ) \left ( W_{h,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes V_h^R\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(h,k) \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes \left ( V_{hk}^R\right ) \right ) \left ( W_{g,1}\right )^*\right )\notag\\ & \quad W_{g,1} \left ( \mathbb I_{{\mathcal H}_L}\otimes \left ( V_{hk}^R\right )^*\right ) \left ( W_{g,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(g,hk) \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes \left ( V_{ghk}^R\right ) \right ) \left ( \mathbb I_{{\mathcal H}_L}\otimes \left ( V_{ghk}^R\right )^*\right )\notag\\ &= c_{R,1}(g,h,k) \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \cdot\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(h,k) \right ) \right )\cdot \left ( W_{g,1}\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right )^* \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(g,hk) \right )\notag\\ &= c_{R,1}(g,h,k)\cdot \mathop{\mathrm{Ad}}\nolimits\left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes V_g^R\right ) W_{g,1} \right ) \left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(h,k) \right ) \right )\cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(g,hk) \right )\notag\\ &= c_{R,1}(g,h,k)\cdot \mathop{\mathrm{Ad}}\nolimits\left ( \left ( {V_g^L}^*\otimes \mathbb I_{{\mathcal H}_R}\right ) V_g W_{g,1} \right ) \left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(h,k) \right ) \right )\cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(g,hk) \right )\notag\\ &= c_{R,1}(g,h,k)\cdot \mathop{\mathrm{Ad}}\nolimits\left ( V_g W_{g,1} \right ) \left ( \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(h,k) \right ) \right )\cdot \left ( \mathbb I_{{\mathcal H}_L}\otimes u_{R}^{(2)}(g,hk) \right ) \end{align} $$

In the last line we used formula (2.130) and Lemma 2.3(iii) to remove ${V_g^L}^*$. From this, we see that $c_{R,1}=c_{R,2}$, completing the proof.

Lemma 2.17. Set

(2.135)$$ \begin{align} \omega\in\mathop{\mathcal{SL}}\nolimits, \quad 0<\theta_1<\theta_2<\frac\pi 2,\quad \text{with } \mathop{\mathrm{IG}}\nolimits(\omega,\theta_1),\ \mathop{\mathrm{IG}}\nolimits(\omega,\theta_2)\neq\emptyset. \end{align} $$

Then we have

(2.136)$$ \begin{align} h^{(5)} \left ( \omega , \theta_1 \right ) = h^{(5)} \left ( \omega, \theta_2 \right ). \end{align} $$

Definition 2.18. From this lemma, for any $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $, we may define

(2.137)$$ \begin{align} h \left ( \omega \right ) := h^{(5)} \left ( \omega, \theta \right ) \end{align} $$

independent of the choice of $\theta $. This is the index we associate to $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $.

Proof. By the assumption, there are some $\left (\tilde \beta _g\right )\in \mathop {\mathrm {IG}}\nolimits (\omega ,\theta _1)$ and $(\eta _g^\sigma )\in {\mathcal T}\left ( \left (\theta _1, \tilde \beta _g\right )\right )$. Because $\omega \in \mathop {\mathcal {SL}}\nolimits $, there are $ \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$ and $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta _2}$ by the definition. Setting $\alpha _0:=\alpha _L\otimes \alpha _R$, we have $\alpha =({\textrm {{inner}}})\circ \alpha _0\circ \Theta $. Because $0<\theta _1<\theta _2<\frac \pi 2$, we also have $(\eta _g^\sigma )\in {\mathcal T}( (\theta _2, \tilde \beta _g ))$, and $\left (\tilde \beta _g\right )\in \mathop {\mathrm {IG}}\nolimits (\omega ,\theta _2)$. For the same reason, we also have $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta _1}$.

By Lemma 2.1, there is some

(2.138)$$ \begin{align} ( (W_g), (u_\sigma(g,h))) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta_1, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ). \end{align} $$

However, we also have

(2.139)$$ \begin{align} ( (W_g), (u_\sigma(g,h))) \in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta_2, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ). \end{align} $$

Therefore, we obtain $h^{(5)} \left ( \omega , \theta _1 \right ) = h^{(5)} \left ( \omega , \theta _2 \right ) $.

This completes the proof of Theorem 2.7.

3 The existence of $\tilde \beta $ for SPT phases

In this section, we give a sufficient condition for $\mathop {\mathrm {IG}}\nolimits (\omega )$ to be nonempty. We consider the same setting as in Section 2.2.

Theorem 3.1. For any $0<\theta <\frac \pi 2$ and $\alpha \in \mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$ satisfying $ \omega _0\circ \alpha \circ \beta _g=\omega _0\circ \alpha $ for all $g\in G$, $\mathop {\mathrm {IG}}\nolimits (\omega _0\circ \alpha ,\theta )$ is not empty.

The strategy is as follows. Our infinite tensor product state $\omega _{0}$ can be written as $\omega _{0}=\omega _{U}\otimes \omega _{D}$, with pure states $\omega _{U}, \omega _{D}$ on ${\mathcal A}_{H_{U}}, {\mathcal A}_{H_{D}}$. Using the factorisation property of $\alpha \in \mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$, we can show that

(3.1)$$ \begin{align} \alpha\circ\beta_{g}^{U}\circ\alpha^{-1} &=\left ( id_{{\mathcal A}_{H_{D}}}\otimes \tilde Y_{g,U}\right )\circ\left ( {\text{an automorphism localised at } C_{\theta_2}} \right )\circ({\textrm{{inner}}}), \end{align} $$

(3.2)$$ \begin{align} \alpha\circ\beta_{g}\circ\alpha^{-1} &=\left ( \tilde Y_{g,D}\otimes \tilde Y_{g,U}\right )\circ\left ( {\text{an automorphism localised at } C_{\theta_2}} \right )\circ({\textrm{{inner}}}), \end{align} $$

with $\tilde Y_{g,U}:=\beta _{g}^{C_{\left (\theta _{0.8},\frac \pi 2\right ],U}}\xi _U, \tilde Y_{g,D}:=\beta _{g}^{C_{\left (\theta _{0.8},\frac \pi 2\right ],U}}\xi _D$ automorphisms on $\left ( C_{\theta _{0.8}}\right )^c\cap H_{U}, \left ( C_{\theta _{0.8}}\right )^c\cap H_{D}$, respectively. The ‘automorphism localised at $C_{\theta _2}$’ can be split into left and right parts. (See equation (3.28).) From the latter equation and $\omega _0\circ \alpha \circ \beta _g=\omega _0\circ \alpha $, one can show that $\omega _{U} \tilde Y_{g,U}$ is quasiequivalent to a state of the form $\varphi _{L}\otimes \varphi _{R}\otimes \omega _{C_{\theta _{2}}^{c}}$, where $\varphi _{L}, \varphi _{R}$ are states on ${\mathcal A}_{C_{\theta _{2}\cap H_{L}}}, {\mathcal A}_{C_{\theta _{2}\cap H_{R}}}$ and $\omega _{C_{\theta _{2}}^{c}}$ is the pure state given as the restriction of $\omega _{0}$ to ${\mathcal A}_{{C_{\theta _{2}}^{c}}}$ (with $\theta _{0.8}<\theta _{2}$). A general lemma proven in the following (Lemma 3.2), derived from the homogeneity of pure state spaces on UHF-algebras, then allows us to show the existence of automorphisms $Z_{g,L}, Z_{g,R}$ on ${\mathcal A}_{C_{\theta _{2}\cap H_{L}}}, {\mathcal A}_{C_{\theta _{2}\cap H_{R}}}$ satisfying $\omega _{U} \tilde Y_{g,U}\simeq \omega _{U}\circ \left ( Z_{g,L}\otimes Z_{g,R}\otimes \mathop {\mathrm {id}}\nolimits _{C_{\theta _{2}}^{c}}\right )$. Combining this with equation (3.1) basically gives the Theorem.

Now let us start with a precise mathematical proof. We first prepare the general lemma just mentioned.

Lemma 3.2. Let ${\mathfrak A},{\mathfrak B}$ be UHF-algebras. Let $\omega $ be a pure state on ${\mathfrak A}\otimes {\mathfrak B}$ and $\varphi _{{\mathfrak A}}, \varphi _{{\mathfrak B}}$ states on ${\mathfrak A}, {\mathfrak B}$, respectively. Assume that $\omega $ is quasiequivalent to $\varphi _{{\mathfrak A}}\otimes \varphi _{{\mathfrak B}}$. Then for any pure states $\psi _{{\mathfrak A}}, \psi _{{\mathfrak B}}$ on ${\mathfrak A}, {\mathfrak B}$, there are automorphisms $\gamma _{{\mathfrak A}}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathfrak A}\right ), \gamma _{{\mathfrak B}}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathfrak B}\right )$ and a unitary $u\in {\mathcal U}\left ({\mathfrak A}\otimes {\mathfrak B}\right )$ such that

(3.3)$$ \begin{align} \omega=\left ( \left ( \psi_{{\mathfrak A}}\circ\gamma_{{\mathfrak A}}\right )\otimes\left ( \psi_{{\mathfrak B}}\circ\gamma_{{\mathfrak B}} \right ) \right )\circ\mathop{\mathrm{Ad}}\nolimits(u). \end{align} $$

If $\psi _{{\mathfrak A}}$ and $\varphi _{{\mathfrak A}}$ are quasiequivalent, then we may set $\gamma _{{\mathfrak A}}=\mathop {\mathrm {id}}\nolimits _{{\mathfrak A}}$.

Proof. Let $({\mathcal H}_{\omega },\pi _{\omega },\Omega _{\omega }), \left ({\mathcal H}_{\varphi _{{\mathfrak A}}},\pi _{\varphi _{{\mathfrak A}}},\Omega _{\varphi _{{\mathfrak A}}}\right ), \left ({\mathcal H}_{\varphi _{{\mathfrak B}}},\pi _{\varphi _{{\mathfrak B}}},\Omega _{\varphi _{{\mathfrak B}}}\right )$ be GNS triples of $\omega , \varphi _{{\mathfrak A}}, \varphi _{{\mathfrak B}}$, respectively. Then $\left ({\mathcal H}_{\varphi _{{\mathfrak A}}}\otimes {\mathcal H}_{\varphi _{{\mathfrak B}}},\pi _{\varphi _{{\mathfrak A}}}\otimes \pi _{\varphi _{{\mathfrak B}}},\Omega _{\varphi _{{\mathfrak A}}}\otimes \Omega _{\varphi _{{\mathfrak B}}}\right )$ is a GNS triple of $\varphi _{{\mathfrak A}}\otimes \varphi _{{\mathfrak B}}$. As $\omega $ is quasiequivalent to $\varphi _{{\mathfrak A}}\otimes \varphi _{{\mathfrak B}}$, there is a $*$-isomorphism $\tau :\pi _{\omega }\left ({\mathfrak A}\otimes {\mathfrak B}\right )''\to \pi _{\varphi _{{\mathfrak A}}}({\mathfrak A})''\otimes \pi _{\varphi _{{\mathfrak B}}}({\mathfrak B})''$ such that $\tau \circ \pi _{\omega }=\pi _{\varphi _{{\mathfrak A}}}\otimes \pi _{\varphi _{{\mathfrak B}}}$. Because $\omega $ is pure, we have $\pi _{\omega }\left ({\mathfrak A}\otimes {\mathfrak B}\right )''={\mathcal B}({\mathcal H}_{\omega })$, and from the isomorphism $\tau $, we have that $\pi _{\varphi _{{\mathfrak A}}}({\mathfrak A})''\otimes \pi _{\varphi _{{\mathfrak B}}}({\mathfrak B})''$ is also a type I factor. Then from [T, Theorem 2.30V], both $\pi _{\varphi _{{\mathfrak A}}}({\mathfrak A})''$ and $\pi _{\varphi _{{\mathfrak B}}}({\mathfrak B})''$ are type I factors. The restriction of $\tau $ to $\pi _{\omega }\left ({\mathfrak A}\otimes {\mathbb C}\mathbb I_{{\mathfrak B}}\right )''$ implies a $*$-isomorphism from $\pi _{\omega }\left ({\mathfrak A}\otimes {\mathbb C}\mathbb I_{{\mathfrak B}}\right )''$ onto the type I factor $\pi _{\varphi _{{\mathfrak A}}}({\mathfrak A})''$. Hence we see that $\pi _{\omega }\left ({\mathfrak A}\otimes {\mathbb C}\mathbb I_{{\mathfrak B}}\right )''$ is a type I factor. Therefore, from [T, Theorem 1.31V], there are Hilbert spaces ${\mathcal K}_{{\mathfrak A}},{\mathcal K}_{{\mathfrak B}}$ and a unitary $W: {\mathcal H}_\omega \to {\mathcal K}_{{\mathfrak A}}\otimes {\mathcal K}_{{\mathfrak B}}$ such that $\mathop {\mathrm {Ad}}\nolimits \left ( W\right )\left ( \pi _{\omega }\left ({\mathfrak A}\otimes {\mathbb C}\mathbb I_{{\mathfrak B}}\right )''\right )= {\mathcal B}\left ( {\mathcal K}_{{\mathfrak A}}\right )\otimes {\mathbb C}\mathbb I_{{\mathcal K}_{{\mathfrak B}}} $. Because $\omega $ is pure, we also have $\mathop {\mathrm {Ad}}\nolimits \left ( W\right )\left ( \pi _{\omega }\left ({\mathbb C}\mathbb I_{{\mathfrak A}} \otimes {\mathfrak B}\right )''\right )={\mathbb C}\mathbb I_{{\mathcal K}_{{\mathfrak A}}}\otimes {\mathcal B}({\mathcal K}_{{\mathfrak B}})$. From this, we see that there are irreducible representations $\rho _{{\mathfrak A}}, \rho _{{\mathfrak B}}$ of ${\mathfrak A}$ and ${\mathfrak B}$ on ${\mathcal K}_{{\mathfrak A}}, {\mathcal K}_{{\mathfrak B}}$ such that $\mathop {\mathrm {Ad}}\nolimits (W)\circ \pi _{\omega }=\rho _{{\mathfrak A}}\otimes \rho _{{\mathfrak B}}$. Fix some unit vectors $\xi _{{\mathfrak A}}\in {\mathcal K}_{{\mathfrak A}}, \xi _{{\mathfrak B}}\in {\mathcal K}_{{\mathfrak B}}$. Then because of the irreducibility of $\rho _{{\mathfrak B}}$ and $\rho _{{\mathfrak B}}$, we have that $\omega _{{\mathfrak A}}:=\left \langle \xi _{{\mathfrak A}},\rho _{{\mathfrak A}}\left (\cdot \right )\xi _{{\mathfrak A}}\right \rangle $ and $\omega _{{\mathfrak B}}:=\left \langle \xi _{{\mathfrak B}},\rho _{{\mathfrak B}}\left (\cdot \right )\xi _{{\mathfrak B}}\right \rangle $ are pure states on ${\mathfrak A}, {\mathfrak B}$. By [KOS, Theorem 1.1] (originally proved by Powers [P] for UHF-algebras), for any pure states $\psi _{{\mathfrak A}}, \psi _{{\mathfrak B}}$ on ${\mathfrak A}, {\mathfrak B}$, there exist automorphisms $\gamma _{{\mathfrak A}}\in \mathop {\mathrm {Aut}}\nolimits ({\mathfrak A}), \gamma _{{\mathfrak B}}\in \mathop {\mathrm {Aut}}\nolimits ({\mathfrak B})$ such that $\omega _{{\mathfrak A}}=\psi _{{\mathfrak A}}\circ \gamma _{{\mathfrak A}}$ $\omega _{{\mathfrak B}}=\psi _{{\mathfrak B}}\circ \gamma _{{\mathfrak B}}$. Now for unit vectors $W^{*}\left (\xi _{{\mathfrak A}}\otimes \xi _{{\mathfrak B}}\right ),\Omega _{\omega }\in {\mathcal H}_{\omega }$, by Kadison’s transitivity theorem and the irreducibility of $\pi _\omega $ there exists a unitary $u\in {\mathcal U}\left ( {\mathfrak A}\otimes {\mathfrak B}\right )$ such that $\pi _{\omega }(u)\Omega _{\omega }=W^{*}\left (\xi _{{\mathfrak A}}\otimes \xi _{{\mathfrak B}}\right )$. Substituting this, we obtain

(3.4)$$ \begin{align} \omega&=\left\langle\Omega_{\omega},\pi_{\omega}\left (\cdot\right ) \Omega_{\omega}\right\rangle =\left\langle\pi_{\omega}(u^{*})W^{*}\left (\xi_{{\mathfrak A}}\otimes \xi_{{\mathfrak B}}\right ),\pi_{\omega}\left (\cdot\right )\pi_{\omega}(u^{*})W^{*}\left (\xi_{{\mathfrak A}}\otimes \xi_{{\mathfrak B}}\right )\right\rangle\notag\\ &=\left\langle W^{*}\left (\xi_{{\mathfrak A}}\otimes \xi_{{\mathfrak B}}\right ),\pi_{\omega}\circ\mathop{\mathrm{Ad}}\nolimits(u)\left (\cdot\right ) W^{*}\left (\xi_{{\mathfrak A}}\otimes \xi_{{\mathfrak B}}\right )\right\rangle\notag\\ &=\left\langle\left (\xi_{{\mathfrak A}}\otimes \xi_{{\mathfrak B}}\right ),\left (\rho_{{\mathfrak A}}\otimes \rho_{{\mathfrak B}}\right )\circ\mathop{\mathrm{Ad}}\nolimits(u)\left (\cdot\right ) \left (\xi_{{\mathfrak A}}\otimes \xi_{{\mathfrak B}}\right )\right\rangle \notag \\ &=\left ( \omega_{{\mathfrak A}}\otimes\omega_{{\mathfrak B}}\right )\circ \mathop{\mathrm{Ad}}\nolimits(u) =\left ( \psi_{{\mathfrak A}}\circ \gamma_{{\mathfrak A}}\otimes \psi_{{\mathfrak B}}\circ \gamma_{{\mathfrak B}}\right ) \circ \mathop{\mathrm{Ad}}\nolimits(u). \end{align} $$

Now assume that $\psi _{{\mathfrak A}}$ and $\varphi _{{\mathfrak A}}$ are quasiequivalent – that is, the GNS representations of $\psi _{{\mathfrak A}}, \varphi _{{\mathfrak A}}$, denoted by $\pi _{\psi _{{\mathfrak A}}}$ and $\pi _{\varphi _{{\mathfrak A}}}$ are quasiequivalent. From the foregoing argument, $\pi _{\omega }\rvert _{{\mathfrak A}}$ and $\pi _{\varphi _{{\mathfrak A}}}$ are quasiequivalent. At the same time, $\pi _{\omega }\rvert _{{\mathfrak A}}$ and $\rho _{{\mathfrak A}}$ are quasiequivalent. Therefore, $\pi _{\psi _{{\mathfrak A}}}$ and $\rho _{{\mathfrak A}}$ are quasiequivalent. Because both of them are irreducible, we see that a pure state ${\psi _{{\mathfrak A}}}$ can be represented by a unit vector $\zeta \in {\mathcal K}_{{\mathfrak A}}$, as ${\psi _{{\mathfrak A}}}=\left \langle \zeta ,\rho _{{\mathfrak A}}\left (\cdot \right ) \zeta \right \rangle $. Because $\rho _{{\mathfrak A}}$ is irreducible, by Kadison’s transitivity theorem there exists a unitary $w\in {\mathcal U}\left ( {\mathfrak A}\right )$ such that $\rho _{{\mathfrak A}}(w^{*})\zeta =\xi _{{\mathfrak A}}$. Hence we obtain ${\psi _{{\mathfrak A}}}\circ \mathop {\mathrm {Ad}}\nolimits (w)=\omega _{{\mathfrak A}}$. Substituting this instead of $\omega _{{\mathfrak A}}=\psi _{{\mathfrak A}}\circ \gamma _{{\mathfrak A}}$ in equation (3.4), we obtain

(3.5)$$ \begin{align} \omega= \left ( \psi_{{\mathfrak A}}\otimes \psi_{{\mathfrak B}}\circ \gamma_{{\mathfrak B}}\right ) \circ \mathop{\mathrm{Ad}}\nolimits\left ( \left ( w\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak B}}\right ) u\right ), \end{align} $$

proving the last claim.

Lemma 3.3. Let ${\mathfrak B},{\mathfrak A}_{1,L},{\mathfrak A}_{2,L},{\mathfrak A}_{1,R},{\mathfrak A}_{2,R}$ be UHF-algebras. Set ${\mathfrak A}_{1}:={\mathfrak A}_{1,L}\otimes {\mathfrak A}_{1,R}$, ${\mathfrak A}_{2}:={\mathfrak A}_{2,L}\otimes {\mathfrak A}_{2,R}$, ${\mathfrak A}_{L}:={\mathfrak A}_{1,L}\otimes {\mathfrak A}_{2,L}$ and ${\mathfrak A}_{R}:={\mathfrak A}_{1,R}\otimes {\mathfrak A}_{2,R}$. Let $\omega , \varphi _{L}^{\left (1,2\right )}, \varphi _{R}^{\left (1,2\right )}, \psi $ be pure states on ${\mathfrak B}\otimes {\mathfrak A}_{1}, {\mathfrak A}_{L}, {\mathfrak A}_{R}, {\mathfrak B}$, respectively. Suppose that $\omega $ is quasiequivalent to $\left .\left ( \psi \otimes \varphi _{L}^{\left (1,2\right )}\otimes \varphi _{R}^{\left (1,2\right )} \right )\right \rvert _{{\mathfrak B}\otimes {\mathfrak A}_{1}}$. Then for any pure states $\varphi _{L}^{(1)}, \varphi _{R}^{(1)}$ on ${\mathfrak A}_{1,L}, {\mathfrak A}_{1,R}$, respectively, there are automorphisms $\gamma _{L}^{(1)}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathfrak A}_{1,L}\right ), \gamma _{R}^{(1)}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathfrak A}_{1,R}\right )$ and a unitary $u\in {\mathcal U}\left ({\mathfrak B}\otimes {\mathfrak A}_{1}\right )$ such that

(3.6)$$ \begin{align} \omega=\left ( \psi\otimes \left ( \varphi_{L}^{(1)}\circ\gamma_{L}^{(1)}\right ) \otimes \left ( \varphi_{R}^{(1)}\circ\gamma_{R}^{(1)}\right ) \right )\circ\mathop{\mathrm{Ad}}\nolimits u. \end{align} $$

Proof. Because the pure state $\omega $ is quasiequivalent to $\left .\left ( \psi \otimes \varphi _{L}^{\left (1,2\right )}\otimes \varphi _{R}^{\left (1,2\right )} \right )\right \rvert _{{\mathfrak B}\otimes {\mathfrak A}_{1}} =\psi \otimes \left .\left ( \varphi _{L}^{\left (1,2\right )}\otimes \varphi _{R}^{\left (1,2\right )} \right )\right \rvert _{{\mathfrak A}_{1}}$, applying Lemma 3.2 means that for any pure states $\varphi _{L}^{(1)}, \varphi _{R}^{(1)}$ on ${\mathfrak A}_{1,L}, {\mathfrak A}_{1,R}$, there exist an automorphism $S\in \mathop {\mathrm {Aut}}\nolimits {\mathfrak A}_1$ and a unitary $v\in {\mathcal U}\left ( {\mathfrak B}\otimes {\mathfrak A}_1\right )$ such that

(3.7)$$ \begin{align} \omega=\left ( \psi\otimes\left ( \left ( \varphi_{L}^{(1)} \otimes \varphi_{R}^{(1)}\right )\circ S\right ) \right )\circ \mathop{\mathrm{Ad}}\nolimits v. \end{align} $$

From equation (3.7) and $\omega \sim _{\text {q.e.}}\left .\left ( \psi \otimes \varphi _{L}^{\left (1,2\right )}\otimes \varphi _{R}^{\left (1,2\right )} \right )\right \rvert _{{\mathfrak B}\otimes {\mathfrak A}_{1}}$, we get $ \left ( \psi \otimes \left ( \left ( \varphi _{L}^{(1)} \otimes \varphi _{R}^{(1)}\right )\circ S\right ) \right )\sim _{\text {q.e.}} \left .\left ( \psi \otimes \varphi _{L}^{\left (1,2\right )}\otimes \varphi _{R}^{\left (1,2\right )} \right )\right \rvert _{{\mathfrak B}\otimes {\mathfrak A}_{1}}$, which implies

(3.8)$$ \begin{align} \left ( \varphi_{L}^{(1)} \otimes \varphi_{R}^{(1)}\right )\circ S\sim_{\text{q.e.}} \left.\left ( \varphi_{L}^{\left(1,2\right)}\otimes \varphi_{R}^{\left(1,2\right)} \right )\right\rvert_{{\mathfrak A}_{1}}. \end{align} $$

Applying Lemma 3.2 to formula (3.8), there are automorphisms $\gamma _{L}^{(1)}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathfrak A}_{1,L}\right ), \gamma _{R}^{(1)}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathfrak A}_{1,R}\right )$ and a unitary $w\in {\mathcal U}\left ( {\mathfrak A}_{1}\right )$ such that

(3.9)$$ \begin{align} \left ( \varphi_{L}^{(1)} \otimes \varphi_{R}^{(1)}\right )\circ S=\left (\left ( \varphi_{L}^{(1)}\circ\gamma_{L}^{(1)}\right ) \otimes \left ( \varphi_{R}^{(1)}\circ\gamma_{R}^{(1)}\right ) \right )\circ\mathop{\mathrm{Ad}}\nolimits w. \end{align} $$

Substituting this into equation (3.7), we obtain equation (3.6).

Lemma 3.4. Let ${\mathfrak A}_L, {\mathfrak A}_R, {\mathfrak B}_{LU}, {\mathfrak B}_{LD}, {\mathfrak B}_{RU}, {\mathfrak B}_{RD}, {\mathfrak C}_U, {\mathfrak C}_D$ be UHF-algebras, and set

(3.10)$$ \begin{align} {\mathfrak B}_U&:={\mathfrak B}_{LU}\otimes{\mathfrak B}_{RU},& {\mathfrak B}_D&:={\mathfrak B}_{LD}\otimes{\mathfrak B}_{RD},& {\mathfrak B}_L&:={\mathfrak B}_{LD}\otimes{\mathfrak B}_{LU},& {\mathfrak B}_R&:={\mathfrak B}_{RD}\otimes{\mathfrak B}_{RU}, \notag\\ {\mathfrak A}&:={\mathfrak A}_L\otimes{\mathfrak A}_R,&{\mathfrak B}&:={\mathfrak B}_D\otimes{\mathfrak B}_U={\mathfrak B}_L\otimes{\mathfrak B}_R,& {\mathfrak C}&:={\mathfrak C}_D\otimes{\mathfrak C}_U,& {\mathfrak D}&:={\mathfrak A}\otimes{\mathfrak B}\otimes{\mathfrak C}. \end{align} $$

Let $\omega _X$ be a pure state on each $X={\mathfrak A}_L, {\mathfrak A}_R, {\mathfrak B}_{LU}, {\mathfrak B}_{LD}, {\mathfrak B}_{RU}, {\mathfrak B}_{RD}, {\mathfrak C}_U, {\mathfrak C}_D$, and set

(3.11)$$ \begin{align} \omega_{{\mathfrak B}{\mathfrak C}}^U&:=\omega_{{\mathfrak B}_{LU}}\otimes\omega_{{\mathfrak B}_{RU}}\otimes\omega_{{\mathfrak C}_U}\quad \text{on } {\mathfrak B}_U\otimes{\mathfrak C}_U,\notag\\ \omega_{{\mathfrak B}{\mathfrak C}}^D&:=\omega_{{\mathfrak B}_{LD}}\otimes\omega_{{\mathfrak B}_{RD}}\otimes\omega_{{\mathfrak C}_D}\quad \text{on } {\mathfrak B}_D\otimes{\mathfrak C}_D,\notag\\ \omega_{{\mathfrak A}}&:=\omega_{{\mathfrak A}_L}\otimes\omega_{{\mathfrak A}_R} \quad \text{on }{\mathfrak A},\notag\\ \omega_{{\mathfrak A}{\mathfrak B}}^L &:=\omega_{{\mathfrak A}_L}\otimes\omega_{{\mathfrak B}_{LU}}\otimes\omega_{{\mathfrak B}_{LD}} \quad \text{on } {\mathfrak A}_L\otimes{\mathfrak B}_L, \notag\\ \omega_{{\mathfrak A}{\mathfrak B}}^R &:=\omega_{{\mathfrak A}_R}\otimes\omega_{{\mathfrak B}_{RU}}\otimes\omega_{{\mathfrak B}_{RD}} \quad \text{on } {\mathfrak A}_R\otimes{\mathfrak B}_R, \notag\\ \omega_0&:=\bigotimes_{\substack{X={\mathfrak A}_L, {\mathfrak A}_R, {\mathfrak B}_{LU}, {\mathfrak B}_{LD},\\ {\mathfrak B}_{RU}, {\mathfrak B}_{RD}, {\mathfrak C}_U, {\mathfrak C}_D}}\omega_X\quad \text{on } {\mathfrak D}. \end{align} $$

Let $\alpha ,\hat \alpha $ be automorphisms on ${\mathfrak D}$ which allow the following decompositions:

(3.12)$$ \begin{align} \hat\alpha&=\left (\rho_{{\mathfrak B}{\mathfrak C}}^U\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak A}}\otimes \rho_{{\mathfrak B}{\mathfrak C}}^D\right ) \circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_U}\otimes\hat\gamma_{{\mathfrak A}{\mathfrak B}}^L\otimes \hat\gamma_{{\mathfrak A}{\mathfrak B}}^R \otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_D} \right )\circ({\textrm{{inner}}}), \end{align} $$

(3.13)$$ \begin{align} \alpha&=\left (\rho_{{\mathfrak B}{\mathfrak C}}^U\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak A}}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak B}_D\otimes{\mathfrak C}_D}\right ) \circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_U}\otimes\gamma_{{\mathfrak A}{\mathfrak B}}^L\otimes \gamma_{{\mathfrak A}{\mathfrak B}}^R \otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_D} \right )\circ({\textrm{{inner}}}). \end{align} $$

Here, $\rho _{{\mathfrak B}{\mathfrak C}}^U, \rho _{{\mathfrak B}{\mathfrak C}}^D$ are automorphisms on ${\mathfrak B}_U\otimes {\mathfrak C}_U, {\mathfrak B}_D\otimes {\mathfrak C}_D$, respectively. For each $\sigma =L,R$, $\gamma _{{\mathfrak A}{\mathfrak B}}^{\sigma },\hat \gamma _{{\mathfrak A}{\mathfrak B}}^{\sigma }$ are automorphisms on ${\mathfrak A}_{\sigma }\otimes {\mathfrak B}_{{\sigma }D}\otimes {\mathfrak B}_{{\sigma }U}$. Suppose that $\omega _0\circ \hat \alpha =\omega _0$. Then there are automorphisms $\eta _L,\eta _R$ on ${\mathfrak A}_L\otimes {\mathfrak B}_{LD}\otimes {\mathfrak B}_{LU}, {\mathfrak A}_R\otimes {\mathfrak B}_{RD}\otimes {\mathfrak B}_{RU}$ such that $\omega _0\circ \alpha $ is quasiequivalent to $\omega _0\circ \left (\mathop {\mathrm {id}}\nolimits _{{\mathfrak C}_U}\otimes \eta _L\otimes \eta _R\otimes \mathop {\mathrm {id}}\nolimits _{{\mathfrak C}_D}\right )$.

Proof. First we claim that there are automorphisms $\theta _{\mathfrak B}^{LU}\in \mathop {\mathrm {Aut}}\nolimits {{\mathfrak B}_{LU}}, \theta _{\mathfrak B}^{RU}\in \mathop {\mathrm {Aut}}\nolimits {\mathfrak B}_{RU}$ and a unitary $u\in {\mathcal U}\left ( {\mathfrak B}^U\otimes {\mathfrak C}^U\right )$ such that

(3.14)$$ \begin{align} \omega_{{\mathfrak B}{\mathfrak C}}^U\circ\rho_{{\mathfrak B}{\mathfrak C}}^U =\omega_{{\mathfrak B}{\mathfrak C}}^U\circ \left ( \theta_{{\mathfrak B}}^{LU}\otimes \theta_{{\mathfrak B}}^{RU}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak C}^U}\right ) \circ\mathop{\mathrm{Ad}}\nolimits\left ( u\right ). \end{align} $$

To prove this, we first note that from $\omega _0\circ \hat \alpha =\omega _0$ and the decomposition (3.12), we have

(3.15)$$ \begin{align} \omega_{{\mathfrak B}{\mathfrak C}}^U\circ\rho_{{\mathfrak B}{\mathfrak C}}^U\otimes \omega_{{\mathfrak A}} \otimes \omega_{{\mathfrak B}{\mathfrak C}}^D\circ\rho_{{\mathfrak B}{\mathfrak C}}^D \sim_{\text{q.e.}} \omega_{{\mathfrak C}_U} \otimes\omega_{{\mathfrak A}{\mathfrak B}}^L\circ\left (\widehat{ \gamma_{{\mathfrak A}{\mathfrak B}}^L}\right )^{-1} \otimes \omega_{{\mathfrak A}{\mathfrak B}}^R\circ\left ( \widehat{\gamma_{{\mathfrak A}{\mathfrak B}}^R}\right )^{-1} \otimes\omega_{{\mathfrak C}_D}. \end{align} $$

From this, because both states are pure (hence the restrictions of their GNS representations onto ${\mathfrak C}_U\otimes {\mathfrak B}_U$ are factors), we have

(3.16)$$ \begin{align} \omega_{{\mathfrak B}{\mathfrak C}}^U\circ\rho_{{\mathfrak B}{\mathfrak C}}^U&= \left.\left ( \omega_{{\mathfrak B}{\mathfrak C}}^U\circ\rho_{{\mathfrak B}{\mathfrak C}}^U\otimes \omega_{{\mathfrak A}} \otimes \omega_{{\mathfrak B}{\mathfrak C}}^D\circ\rho_{{\mathfrak B}{\mathfrak C}}^D \right ) \right\rvert_{{\mathfrak C}_U\otimes {\mathfrak B}_U}\notag\\ &\sim_{\text{q.e.}} \omega_{{\mathfrak C}_U} \otimes\left.\left ( \omega_{{\mathfrak A}{\mathfrak B}}^L\circ\left ( \widehat{\gamma_{{\mathfrak A}{\mathfrak B}}^L}\right )^{-1} \otimes \omega_{{\mathfrak A}{\mathfrak B}}^R\circ\left ( \widehat{\gamma_{{\mathfrak A}{\mathfrak B}}^R}\right )^{-1}\right )\right\rvert_{{\mathfrak B}_U}. \end{align} $$

We apply Lemma 3.3 for ${\mathfrak B}, {\mathfrak A}_{1L}, {\mathfrak A}_{1R}, {\mathfrak A}_{2L}, {\mathfrak A}_{2R}, \omega , \varphi _L^{(1,2)}, \varphi _R^{(1,2)}, \psi $, replaced by ${\mathfrak C}_U, {\mathfrak B}_{LU}, {\mathfrak B}_{RU}, {\mathfrak A}_{L}\otimes {\mathfrak B}_{LD}, {\mathfrak A}_R\otimes {\mathfrak B}_{RD}, \omega _{{\mathfrak B}{\mathfrak C}}^U\circ \rho _{{\mathfrak B}{\mathfrak C}}^U, \omega _{{\mathfrak A}{\mathfrak B}}^L\circ \left (\widehat {\gamma _{{\mathfrak A}{\mathfrak B}}^L}\right )^{-1}, \omega _{{\mathfrak A}{\mathfrak B}}^R\circ \left ( \widehat {\gamma _{{\mathfrak A}{\mathfrak B}}^R}\right )^{-1}, \omega _{{\mathfrak C}_U}$, respectively. From equation (3.16), they satisfy the conditions in Lemma 3.3. Applying Lemma 3.3 – for pure states $\varphi _L^{(1)}=\omega _{{\mathfrak B}_{LU}}$ and $\varphi _R^{(1)}=\omega _{{\mathfrak B}_{RU}}$ – we obtain automorphisms $\theta _{{\mathfrak B}}^{LU}\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathfrak B}_{LU}\right ), \theta _{{\mathfrak B}}^{RU}\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathfrak B}_{RU}\right )$ and a unitary $u\in {\mathcal U}\left ({\mathfrak B}_U\otimes {\mathfrak C}_U\right )$ satisfying equation (3.14).

We set

(3.17)$$ \begin{align} \begin{split} \eta_L&:=\left ( \theta^{LU}_{{\mathfrak B}}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak A}_L}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak B}_{LD}}\right ) \circ\gamma_{{\mathfrak A}{\mathfrak B}}^L \in \mathop{\mathrm{Aut}}\nolimits\left ({\mathfrak B}_{LU}\otimes {\mathfrak A}_L\otimes{\mathfrak B}_{LD}\right )\\ \eta_R&:=\left ( \theta^{RU}_{{\mathfrak B}}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak A}_R}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak B}_{RD}}\right ) \circ\gamma_{{\mathfrak A}{\mathfrak B}}^R \in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathfrak B}_{RU}\otimes {\mathfrak A}_R\otimes{\mathfrak B}_{RD}\right ). \end{split} \end{align} $$

Then we have

(3.18)$$ \begin{align} \begin{split} \omega_0\circ\alpha&= \left ( \omega_{{\mathfrak A}_L}\otimes\omega_{{\mathfrak A}_R} \otimes \omega_{{\mathfrak B}{\mathfrak C}}^U\otimes \omega_{{\mathfrak B}{\mathfrak C}}^D \right )\circ\alpha\\ &\sim_{\text{q.e.}} \left ( \omega_{{\mathfrak A}_L}\otimes\omega_{{\mathfrak A}_R} \otimes \omega_{{\mathfrak B}{\mathfrak C}}^U\circ\rho_{{\mathfrak B}{\mathfrak C}}^U \otimes \omega_{{\mathfrak B}{\mathfrak C}}^D \right ) \circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_U}\otimes \gamma_{{\mathfrak A}{\mathfrak B}}^L\otimes \gamma_{{\mathfrak A}{\mathfrak B}}^R \otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_D} \right )\\ &\sim_{\text{q.e.}} \left ( \omega_{{\mathfrak A}_L}\otimes\omega_{{\mathfrak A}_R} \otimes \omega_{{\mathfrak B}{\mathfrak C}}^U \otimes \omega_{{\mathfrak B}{\mathfrak C}}^D \right )\\ &\quad \circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_U}\otimes \left ( \left ( \theta_{{\mathfrak B}}^{LU}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak A}_L}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak B}_{LD}} \right )\circ\gamma_{{\mathfrak A}{\mathfrak B}}^L\right ) \otimes \left ( \left ( \theta_{{\mathfrak B}}^{RU}\otimes \mathop{\mathrm{id}}\nolimits_{{\mathfrak A}_R}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak B}_{RD}} \right )\circ\gamma_{{\mathfrak A}{\mathfrak B}}^R\right ) \otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_D} \right )\\ &=\omega_0\circ\left (\mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_U}\otimes\eta_L\otimes\eta_R\otimes\mathop{\mathrm{id}}\nolimits_{{\mathfrak C}_D}\right ). \end{split} \end{align} $$

This completes the proof.

Now we are ready to prove Theorem 3.1.

Proof of Theorem 3.1.

Set $0<\theta <\frac \pi 2$ and $\alpha \in \mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$ satisfying $ \omega _0\circ \alpha \circ \beta _g=\omega _0\circ \alpha $ for all $g\in G$. We would like to show that $\mathop {\mathrm {IG}}\nolimits (\omega _0\circ \alpha ,\theta )$ is not empty.

Let us set $\theta _{2.2}:=\theta $ and consider $\theta _{0.8}, \theta _1, \theta _{1.2}, \theta _{1.8}, \theta _2, \theta _{2.8}, \theta _3, \theta _{3.2}$ satisfying formula (2.11) for this $\theta _{2.2}$. Because $\alpha \in \mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$, there is a decomposition given by formulas (2.12), (2.13) and (2.14). Using this decomposition, set

(3.19)$$ \begin{align} \begin{split} \alpha_1&:=\alpha_{1D}\otimes\alpha_{1U},\quad \text{where}\\ \alpha_{1\zeta} &:= \left ( \alpha_{\left(\theta_1,\theta_2\right],\zeta} \otimes \alpha_{\left(\theta_2,\theta_3\right],\zeta}\otimes \alpha_{\left(\theta_3,\frac\pi 2\right],\zeta} \right ) \\ &\quad \circ \left ( \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\zeta}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right],\zeta} \otimes \alpha_{\left(\theta_{2.8},\theta_{3.2}\right],\zeta} \right )\in\mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{\left ( \left ( C_{\theta_{0.8}}\right )^c\right )_\zeta} \right ) ,\quad\zeta=U,D,\\ \alpha_2&:=\alpha_{\left[0,\theta_1\right]} \in\mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{\theta_1}}\right ). \end{split} \end{align} $$

We have $\alpha =({\textrm {{inner}}})\circ \alpha _2\circ \alpha _1$.

We would like to show that $\left (\alpha \circ \beta _g^U\circ \alpha ^{-1}, \alpha \circ \beta _g\circ \alpha ^{-1}\right )$ satisfy the conditions of $(\alpha ,\hat \alpha )$ in Lemma 3.4. We first show that they satisfy a decomposition corresponding to equations (3.12) and (3.13). For $\Gamma ={\mathbb Z}^2, H_U$, we have

(3.20)$$ \begin{align} \left ( \beta_g^\Gamma\right )^{-1}\alpha\circ\beta_g^\Gamma\circ\alpha^{-1} =({\textrm{{inner}}})\circ\left ( \beta_g^\Gamma\right )^{-1}\circ\left ( \alpha_1\beta_g^\Gamma\alpha_1^{-1}\right ) \left ( \alpha_1\beta_g^\Gamma\alpha_1^{-1}\right )^{-1} \alpha_2\alpha_1\beta_g^{\Gamma}\alpha_1^{-1}\alpha_2^{-1}. \end{align} $$

The latter part, $\left ( \alpha _1\beta _g^\Gamma \alpha _1^{-1}\right )^{-1} \alpha _2\alpha _1\beta _g^{\Gamma }\alpha _1^{-1}\alpha _2^{-1}$, decomposes to left and right. To see this, first note that

(3.21)$$ \begin{align} \alpha_1^{-1}\alpha_2\alpha_1 =\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}^{-1}\alpha_{\left[0,\theta_1\right]}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]} \in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{C_{\theta_{1.2}}} \right ). \end{align} $$

Because the conjugation $\left ( \beta _g^\Gamma \right )^{-1}\cdot \beta _g^{\Gamma }$ does not change the support of an automorphism, $\left ( \beta _g^\Gamma \right )^{-1}\left ( \alpha _1^{-1}\alpha _2\alpha _1\right ) \beta _g^{\Gamma }$ is also supported on ${C_{\theta _{1.2}}}$. Therefore, we have

(3.22)$$ \begin{align} &\alpha_1\left ( \left ( \beta_g^\Gamma\right )^{-1}\left ( \alpha_1^{-1}\alpha_2\alpha_1\right ) \beta_g^{\Gamma}\right ) \alpha_1^{-1} \notag \\ &=\alpha_{\left(\theta_1,\theta_2\right]}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]} \left ( \beta_g^\Gamma\right )^{-1} \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}^{-1}\alpha_{\left[0,\theta_1\right]}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]} \beta_g^\Gamma\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}^{-1}\alpha_{\left(\theta_1,\theta_2\right]}^{-1}. \end{align} $$

Hence we get the left-right decomposition

(3.23)$$ \begin{align} \begin{split} \left ( \alpha_1\beta_g^\Gamma\alpha_1^{-1}\right )^{-1} &\alpha_2\alpha_1\beta_g^{\Gamma}\alpha_1^{-1}\alpha_2^{-1} =\alpha_1\left ( \left ( \beta_g^\Gamma\right )^{-1}\left ( \alpha_1^{-1}\alpha_2\alpha_1\right ) \beta_g^{\Gamma}\right ) \alpha_1^{-1}\alpha_2^{-1}\\ &=\alpha_{\left(\theta_1,\theta_2\right]}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]} \left ( \beta_g^\Gamma\right )^{-1} \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}^{-1}\alpha_{\left[0,\theta_1\right]}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]} \beta_g^\Gamma\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}^{-1}\alpha_{\left(\theta_1,\theta_2\right]}^{-1}\circ\alpha_{\left[0,\theta_1\right]}^{-1}\\ &=:\bigotimes_{\sigma=L,R}\Xi_{\Gamma,g,\sigma}. \end{split} \end{align} $$

Here we set

(3.24)$$ \begin{align} &\Xi_{\Gamma,g,\sigma}:=\left ( \alpha_{\left(\theta_1,\theta_2\right],\sigma}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma} \left ( \beta_g^{\Gamma_\sigma}\right )^{-1} \right. \notag\\ &\quad \left.\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma}^{-1}\alpha_{\left[0,\theta_1\right],\sigma }\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma} \beta_g^{\Gamma_\sigma}\alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma}^{-1}\alpha_{\left(\theta_1,\theta_2\right],\sigma}^{-1}\circ\alpha_{\left[0,\theta_1\right],\sigma}^{-1} \right )\quad \in\mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{\left ( C_{\theta_2}\right )_\sigma}\right ). \end{align} $$

On the other hand, the first part of equation (3.20) with $\Gamma ={\mathbb Z}^2,H_U$ satisfies

(3.25)$$ \begin{align} \begin{split} \beta_g^{-1}\alpha_1\beta_g\alpha_1^{-1} = \xi_D\otimes\xi_U, \qquad \left ( \beta_g^{U}\right )^{-1}\alpha_1\beta_g^U\alpha_1^{-1} = \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_D}}\otimes \xi_U, \end{split} \end{align} $$

where

(3.26)$$ \begin{align} \xi_\zeta:=\left ( \beta_g^{\zeta}\right )^{-1}\alpha_{1,\zeta}\beta_g^{\zeta}\alpha_{1,\zeta}^{-1}\in \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{\left ( \left ( C_{\theta_{0.8}}\right )^c\right )_\zeta} \right ),\quad \zeta=U,D. \end{align} $$

Hence we obtain decompositions

(3.27)$$ \begin{align} \begin{split} \left ( \beta_g^U\right )^{-1}\circ\alpha\circ \beta_g^U\circ \alpha^{-1} &=({\textrm{{inner}}})\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_D}}\otimes \xi_U \right )\circ \left ( \Xi_{H_U,g,L}\otimes \Xi_{H_U,g,R}\right ),\\ \left ( \beta_g\right )^{-1}\circ\alpha\circ \beta_g\circ \alpha^{-1} &=({\textrm{{inner}}})\circ \left ( \xi_D\otimes\xi_U\right ) \circ \left ( \Xi_{{\mathbb Z}^2,g,L}\otimes \Xi_{{\mathbb Z}^2,g,R}\right ). \end{split} \end{align} $$

Because $\xi _{\zeta }\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{\left ( \left ( C_{\theta _{0.8}}\right )^c\right )_\zeta } \right )$ commutes with $\beta _{g}^{C_{\left [0,\theta _{0.8}\right ]}}$ and $\beta _{g}^{C_{\left [0,\theta _{0.8}\right ],U}}$, we get

(3.28)$$ \begin{align} \alpha\circ \beta_g^U\circ \alpha^{-1} &=({\textrm{{inner}}})\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{H_D}}\otimes \beta_{g}^{C_{\left(\theta_{0.8},\frac\pi 2\right],U}}\xi_U \right )\circ \left ( \beta_{g}^{C_{\left[0,\theta_{0.8}\right],L,U}}\Xi_{H_U,g,L}\otimes \beta_{g}^{C_{\left[0,\theta_{0.8}\right],R,U}} \Xi_{H_U,g,R}\right ), \notag\\ \alpha\circ \beta_g\circ \alpha^{-1} &=({\textrm{{inner}}})\circ \left ( \beta_{g}^{C_{(\theta_{0.8},\frac\pi 2],D}} \xi_D\otimes\beta_{g}^{C_{(\theta_{0.8},\frac\pi 2],U}}\xi_U \!\right ) \circ \left ( \beta_{g}^{C_{[0,\theta_{0.8}],L}}\Xi_{{\mathbb Z}^2,g,L}\otimes \beta_{g}^{C_{[0,\theta_{0.8}],R}}\Xi_{{\mathbb Z}^2,g,R} \!\right ). \end{align} $$

Furthermore, from the $\beta _g$-invariance of $\omega _0\circ \alpha $, we have

(3.29)$$ \begin{align} \omega_0\circ \alpha\circ \beta_g\circ \alpha^{-1} =\omega_0. \end{align} $$

Now we apply Lemma 3.4 for ${\mathfrak A}_\sigma , {\mathfrak B}_{\sigma \zeta }, {\mathfrak C}_\zeta $ replaced by ${\mathcal A}_{\left ( C_{\left [0,\theta _{0.8}\right ]}\right )_\sigma }, {\mathcal A}_{\left ( C_{\left (\theta _{0.8},\theta _{2}\right ]}\right )_{\sigma ,\zeta }}, {\mathcal A}_{\left ( C_{\left (\theta _2,\frac \pi 2\right ]}\right )_\zeta }$, for $\sigma =L,R$, $\zeta =D,U$. By equations (3.29) and (3.28), $\left (\alpha \circ \beta _g^U\circ \alpha ^{-1}, \alpha \circ \beta _g\circ \alpha ^{-1}\right )$ satisfy the conditions of $(\alpha ,\hat \alpha )$ in Lemma 3.4, for $\omega _0$ and its restrictions. Applying Lemma 3.4, there are $\tilde \eta _{\sigma ,g}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{\left ( C_{\theta _2}\right )_\sigma }\right )$, $g\in G$, $\sigma =L,R$, such that

(3.30)$$ \begin{align} \omega_0\circ\alpha\circ \beta_g^U\circ \alpha^{-1} \sim_{\text{q.e.}} \omega_0\circ \left ( \tilde\eta_{Lg}\otimes \tilde\eta_{Rg} \right ),\quad g\in G. \end{align} $$

Because both $\omega _0\circ \alpha \circ \beta _g^U\circ \alpha ^{-1}$ and $\omega _0\circ \left ( \tilde \eta _{Lg}\otimes \tilde \eta _{Rg} \right )$ are pure, by Kadison’s transitivity theorem there exists a unitary $\tilde v_g\in {\mathcal U}({\mathcal A})$ such that

(3.31)$$ \begin{align} \omega_0\circ\alpha\circ \beta_g^U\circ \alpha^{-1} = \omega_0\circ\mathop{\mathrm{Ad}}\nolimits_{\tilde v_g}\circ \left ( \tilde\eta_{Lg}\otimes \tilde\eta_{Rg} \right ),\quad g\in G. \end{align} $$

We define

(3.32)$$ \begin{align} \tilde \beta_g:= \mathop{\mathrm{Ad}}\nolimits\left ( \alpha^{-1}\left (\tilde v_{g^{-1}}\right )\right ) \circ\alpha^{-1} \circ\left ( \tilde\eta_{Lg^{-1}}\otimes \tilde\eta_{Rg^{-1}} \right )\circ\alpha\circ\beta_g^U,\quad g\in G. \end{align} $$

It suffices to show that $\left (\tilde \beta _g\right )\in \mathop {\mathrm {IG}}\nolimits (\omega _0\circ \alpha ,\theta )=\mathop {\mathrm {IG}}\nolimits \left (\omega _0\circ \alpha ,\theta _{2.2}\right )$. By equation (3.31), we have $\omega _0\circ \alpha \circ \tilde \beta _g=\omega _0\circ \alpha $. Therefore, what is left to be proven is that there are $\eta _{g}^\sigma \in \mathop {\mathrm {Aut}}\nolimits \left ( \left ( C_\theta \right )_\sigma \right )$, $g\in G$, $\sigma =L,R$, such that

(3.33)$$ \begin{align} \tilde\beta_g=({\textrm{{inner}}})\circ\left ( \eta_g^L\otimes\eta_g^R\right )\circ\beta_g^U,\quad \text{for all } g\in G. \end{align} $$

By the decomposition (2.12) and the fact that $\tilde \eta _{Lg^{-1}}\otimes \tilde \eta _{Rg^{-1}}$ has support in $C_{\theta _2}$, we have

(3.34)$$ \begin{align} \begin{split} \alpha^{-1} &\circ\left ( \tilde\eta_{Lg^{-1}}\otimes \tilde\eta_{Rg^{-1}} \right )\circ\alpha\\ &=({\textrm{{inner}}})\circ \left ( \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right]} \right )^{-1} \left ( \alpha_{\left[0,\theta_1\right]}\otimes\alpha_{\left(\theta_1,\theta_2\right]} \right )^{-1} \left ( \tilde\eta_{Lg^{-1}}\otimes \tilde\eta_{Rg^{-1}} \right ) \left ( \alpha_{\left[0,\theta_1\right]}\otimes\alpha_{\left(\theta_1,\theta_2\right]} \right )\\ & \quad \circ \left ( \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right]} \right )\\ &=({\textrm{{inner}}})\circ\left ( \eta_g^L\otimes\eta_g^R\right ), \end{split} \end{align} $$

where

(3.35)$$ \begin{align} \begin{split} \eta_{g}^\sigma&=\left ( \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right],\sigma} \right )^{-1} \left ( \alpha_{\left[0,\theta_1\right],\sigma}\otimes\alpha_{\left(\theta_1,\theta_2\right],\sigma} \right )^{-1} \left ( \tilde\eta_{\sigma g^{-1}} \right ) \left ( \alpha_{\left[0,\theta_1\right],\sigma}\otimes\alpha_{\left(\theta_1,\theta_2\right],\sigma} \right ) \\ & \quad \circ \left ( \alpha_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right], \sigma} \right )\\ &\quad \in \mathop{\mathrm{Aut}}\nolimits\left ( \left ( C_{\theta_{2.2}}\right )_\sigma\right ),\quad \sigma=L,R. \end{split} \end{align} $$

Substituting this into formula (3.32), we obtain equation (3.33). This completes the proof.

4 The stability of the index $h(\omega )$

In this section we prove the stability of the index $h(\omega )$ with respect to $\gamma \in \mathop {\mathrm {GUQAut}}\nolimits ({\mathcal A})$.

Theorem 4.1. Set $\omega \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $. Set $\gamma \in \mathop {\mathrm {GUQAut}}\nolimits \left ({\mathcal A}\right )$. Then we have $\omega \circ \gamma \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega \circ \gamma )\neq \emptyset $ and

(4.1)$$ \begin{align} h(\omega\circ\gamma)=h(\omega). \end{align} $$

Proof. The point of the proof is that we can derive $\left (\hat \alpha _{L},\hat \alpha _{R},\hat \Theta \right )\in {\mathcal D}_{\alpha \gamma }^{\theta _{1.2}}$ (formulas (4.10) and (4.11)) and $\left ( \gamma ^{-1}\tilde \beta _{g}\gamma \right )\in \mathop {\mathrm {IG}}\nolimits (\omega \circ \gamma ,\theta _{1.2}), \left (\hat \eta _{g}^{\sigma }\right )\in {\mathcal T}\left (\theta _{1.2}, \left ( \gamma ^{-1}\tilde \beta _{g}\gamma \right )\right )$ (formula (4.16)) from the corresponding objects for $\alpha $, using the factorisation property of $\alpha ,\gamma $. And it is straightforward to see that the $\beta _{g}^{U}$-invariance of $\gamma _{C}$ results in $\mathop {\mathrm {IP}}\nolimits \left ( \omega , \alpha , \theta _{2}, \left (\tilde \beta _g\right ), (\eta _{g}^\sigma ), (\alpha _L,\alpha _R,\Theta ) \right ) =\mathop {\mathrm {IP}}\nolimits \left ( \omega \circ \gamma , \alpha \circ \gamma , \theta _{1.2}, \left (\gamma ^{-1}\tilde \beta _g\gamma \right ), \left (\hat \eta _{g}^\sigma \right ), \left (\hat \alpha _L,\hat \alpha _R,\hat \Theta \right ) \right )$, which immediately implies the Theorem.

Step 1. From $\omega \in \mathop {\mathcal {SL}}\nolimits $, there is an $ \alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$. For any $0<\theta <\frac \pi 2$ fixed, we show that ${\mathcal D}^{\theta }_{\alpha \circ \gamma }\neq \emptyset $, hence $\alpha \circ \gamma \in \mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$ and $\omega \circ \gamma =\omega _{0}\circ \alpha \gamma \in \mathop {\mathcal {SL}}\nolimits $. Set $\theta _{1.2}:=\theta $ and choose

(4.2)$$ \begin{align} 0<\theta_{0}<\theta_{0.8}<\theta_1<\theta_{1.2}:=\theta<\theta_{1.8}<\theta_2<\theta_{2.2}< \theta_{2.8}<\theta_3<\theta_{3.2}<\frac\pi 2. \end{align} $$

Because $\alpha \in \mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$, there exists some $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta _{2}}$. Setting $\alpha _0:=\alpha _L\otimes \alpha _R$, we have $\alpha =({\textrm {{inner}}})\circ \alpha _0\circ \Theta $. Because $\gamma \in \mathop {\mathrm {GUQAut}}\nolimits \left ({\mathcal A}\right )$, there are $\gamma _{H}\in \mathop {\mathrm {HAut}}\nolimits ({\mathcal A})$ and $\gamma _{C}\in \mathop {\mathrm {GSQAut}}\nolimits ({\mathcal A})$ such that

(4.3)$$ \begin{align} \gamma=\gamma_{C}\circ\gamma_{H}. \end{align} $$

Because $\gamma _{H}\in \mathop {\mathrm {HAut}}\nolimits ({\mathcal A})$, we may decompose $\gamma _{H}$ as

(4.4)$$ \begin{align} \gamma_{H}=({\textrm{{inner}}})\circ\left ( \gamma_{H, {L}}\otimes \gamma_{H,R}\right ) =({\textrm{{inner}}})\circ\gamma_{0}, \end{align} $$

with some $\gamma _{H, {\sigma }}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{{\left ( C_{\theta _{0}}\right )_\sigma }}\right )$, $\sigma =L,R$. We set $\gamma _{0}:=\gamma _{H, {L}}\otimes \gamma _{H,R}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{{ C_{\theta _{0}}}}\right )$. By definition, $\gamma _{C}\in \mathop {\mathrm {GSQAut}}\nolimits ({\mathcal A})$ allows a decomposition

(4.5)$$ \begin{align} \gamma_{C}&=({\textrm{{inner}}})\circ\gamma_{CS},\notag\\ \gamma_{CS}&=\left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \otimes \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ), \end{align} $$

with

(4.6)$$ \begin{align} \gamma_X&:=\bigotimes_{\sigma=L,R, \, \zeta=D,U} \gamma_{X,\sigma,\zeta},& \gamma_{\left[0,\theta_{1}\right]}&:=\bigotimes_{\sigma=L,R}\gamma_{\left[0,\theta_{1}\right],\sigma},& \gamma_{\left(\theta_3,\frac\pi 2\right]}&:=\bigotimes_{\zeta=D,U} \gamma_{\left(\theta_3,\frac\pi 2\right],\zeta}, \nonumber\\ \gamma_{X,\sigma,\zeta}&\in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{X,\sigma,\zeta}}\right ),& \gamma_{X,\sigma}&:=\bigotimes_{\zeta=U,D}\gamma_{X,\sigma,\zeta},& \gamma_{X,\zeta}&:=\bigotimes_{\sigma=L,R}\gamma_{X,\sigma,\zeta},\\ \gamma_{\left[0,\theta_{1}\right],\sigma}&\in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{\left[0,\theta_{1}\right],\sigma}}\right ),& \gamma_{\left(\theta_3,\frac\pi 2\right],\zeta}&\in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{\left(\theta_3,\frac\pi 2\right],\zeta}}\right ), \nonumber \end{align} $$

for

(4.7)$$ \begin{align} X=(\theta_1,\theta_2], (\theta_2,\theta_3], (\theta_{0.8},\theta_{1.2}], (\theta_{1.8},\theta_{2.2}], (\theta_{2.8},\theta_{3.2}],\quad \sigma=L,R,\ \zeta=D,U. \end{align} $$

Here we have

(4.8)$$ \begin{align} \gamma_{I}\circ\beta_g^{U}=\beta_g^{U}\circ\gamma_{I}\quad\text{for all } g\in G, \end{align} $$

for any

(4.9)$$ \begin{align} I=[0,\theta_1],(\theta_1,\theta_2], (\theta_2,\theta_3], \left(\theta_3,\frac\pi 2\right], (\theta_{0.8}, \theta_{1.2}], (\theta_{1.8},\theta_{2.2}], (\theta_{2.8},\theta_{3.2}]. \end{align} $$

Set

(4.10)$$ \begin{align} \hat\Theta:=\Theta\circ \left ( \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ) \in\mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{C_{\theta_{1.8}}^{c}}\right ) \subset \mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{C_{\theta_{1.2}}^{c}}\right ) \end{align} $$

and

(4.11)$$ \begin{align} \hat\alpha_{\sigma}:= \alpha_{\sigma}\circ \left ( \gamma_{\left[0,\theta_1\right],\sigma}\otimes\gamma_{\left(\theta_1,\theta_2\right],\sigma} \right ) \circ \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma} \circ\gamma_{H,\sigma}\in\mathop{\mathrm{Aut}}\nolimits\left({\mathcal A}_{H_{\sigma}}\right),\quad\sigma=L,R. \end{align} $$

We claim

(4.12)$$ \begin{align} \alpha\circ\gamma=({\textrm{{inner}}})\circ\left (\hat\alpha_{L}\otimes\hat \alpha_{R}\right )\circ\hat\Theta. \end{align} $$

This means $(\hat \alpha _{L},\hat \alpha _{R},\hat \Theta )\in {\mathcal D}_{\alpha \gamma }^{\theta _{1.2}}$, hence ${\mathcal D}_{\alpha \gamma }^{\theta }={\mathcal D}_{\alpha \gamma }^{\theta _{1.2}}\neq \emptyset $. The claim (4.12) can be checked as follows. Note that $\gamma _{\left (\theta _2,\theta _3\right ]}\otimes \gamma _{\left (\theta _3,\frac \pi 2\right ]}$ and $\gamma _{\left (\theta _{0.8}, \theta _{1.2}\right ]}$ commute because of their disjoint supports. Because $\Theta \in \mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{C_{\theta _{2}}^{c}})$, it commutes with $\gamma _{\left [0,\theta _1\right ]}\otimes \gamma _{\left (\theta _1,\theta _2\right ]}$ and $\gamma _{\left (\theta _{0.8}, \theta _{1.2}\right ]}$. Therefore, we have

(4.13)$$ \begin{align} \alpha\circ\gamma&=({\textrm{{inner}}})\circ\alpha_{0}\circ \Theta\circ\left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \otimes \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \notag\\ &\quad \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )\circ\gamma_{0}\notag\\ &=({\textrm{{inner}}})\circ\alpha_{0}\circ\left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \right )\circ\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\circ \Theta\circ\left ( \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]}\right ) \notag\\ &\quad \circ\left ( \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )\circ\gamma_{0}\notag\\ &=({\textrm{{inner}}})\circ\alpha_{0}\circ\left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \right )\circ\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\circ\hat\Theta\circ\gamma_{0}. \end{align} $$

Because $\gamma _{0}\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{{ C_{\theta _{0}}}}\right )$ and $\hat \Theta \in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{C_{\theta _{1.8}}^{c}}\right )$ commute, we have

(4.14)$$ \begin{align} \alpha\circ\gamma&=\text{equation }(4.13)= ({\textrm{{inner}}})\circ\alpha_{0}\circ\left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \right )\circ\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\circ\gamma_{0}\circ\hat\Theta \notag\\ &=({\textrm{{inner}}})\circ\left (\hat\alpha_{L}\otimes\hat \alpha_{R}\right )\circ\hat\Theta, \end{align} $$

proving equation (4.12).

Step 2. From $\mathop {\mathrm {IG}}\nolimits (\omega )\neq \emptyset $, we fix a $0<\theta _{0}<\frac \pi 2$ such that $\mathop {\mathrm {IG}}\nolimits (\omega ,\theta _{0})\neq \emptyset $. We choose $\theta _{0.8},\theta _1,\theta _{1.2},\theta _{1.8},\theta _2,\theta _{2.2}, \theta _{2.8},\theta _3,\theta _{3.2}$ such that

(4.15)$$ \begin{align} 0<\theta_{0}<\theta_{0.8}<\theta_1<\theta_{1.2}<\theta_{1.8}<\theta_2<\theta_{2.2}< \theta_{2.8}<\theta_3<\theta_{3.2}<\frac\pi 2. \end{align} $$

For these $\theta $s, we associate the decomposition of $\gamma $ in step 1. Fix $\left ( \tilde \beta _{g}\right )\in \mathop {\mathrm {IG}}\nolimits (\omega ,\theta _{0})$ and $ \left (\eta _{g}^{\sigma }\right )\in {\mathcal T}\left (\theta _{0}, \left (\tilde \beta _g\right )\right )$. Set $\eta _{g}:=\eta _{g}^{L}\otimes \eta _{g}^{R}$. Note that $\left (\eta _{g}^{\sigma }\right )$ also belongs to ${\mathcal T}\left (\theta _{2}, \left (\tilde \beta _g\right )\right )$. Set

(4.16)$$ \begin{align} \hat\eta_{g}^{\sigma} &:= \left ( \gamma_{\left[0,\theta_1\right],\sigma}\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma}\gamma_{H,\sigma}\right )^{-1} \eta_{g}^{\sigma}\left ( \beta_{g}^{\sigma U} \gamma_{\left[0,\theta_1\right],\sigma}\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],\sigma}\gamma_{H,\sigma} \left ( \beta_{g}^{\sigma U} \right )^{-1}\right )\notag\\ &\quad \in\mathop{\mathrm{Aut}}\nolimits\left ( {\mathcal A}_{\left ( C_{\theta_{1.2}}\right )_{\sigma}}\right ), \end{align} $$

for $\sigma =L,R$. We also set $\hat \eta _{g}:=\hat \eta _{g}^{L}\otimes \hat \eta _{g}^{R}$. We claim that $\left ( \gamma ^{-1}\tilde \beta _{g}\gamma \right )\in \mathop {\mathrm {IG}}\nolimits (\omega \circ \gamma ,\theta _{1.2})$ with $\left (\hat \eta _{g}^{\sigma }\right )\in {\mathcal T}\left (\theta _{1.2}, \left ( \gamma ^{-1}\tilde \beta _{g}\gamma \right )\right )$. Clearly we have

(4.17)$$ \begin{align} \omega\circ\gamma\circ\left ( \gamma^{-1}\tilde \beta_{g}\gamma\right ) =\omega\circ\tilde \beta_{g}\circ\gamma =\omega\circ\gamma. \end{align} $$

Therefore, what remains to be shown is

(4.18)$$ \begin{align} \gamma^{-1}\tilde \beta_{g}\gamma =({\textrm{{inner}}}) \circ\left (\hat\eta_{g}^L\otimes \hat\eta_{g}^R\right )\circ \beta_{g}^{U}. \end{align} $$

To see this, we first have

(4.19)$$ \begin{align} &\gamma^{-1}\circ\eta_{g}\circ\gamma=({\textrm{{inner}}})\circ\gamma_{0}^{-1}\circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )^{-1}\notag\\ &\quad \circ \left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \otimes \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right )^{-1}\notag\\ & \quad \circ\eta_{g}\circ\left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \otimes \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ) \gamma_{0} \end{align} $$

from the decomposition of equations (4.3), (4.4) and (4.5). Because $\gamma _{\left (\theta _1,\theta _2\right ]} \otimes \gamma _{\left (\theta _2,\theta _3\right ]}\otimes \gamma _{\left (\theta _3,\frac \pi 2\right ]}$ commutes with $\eta _{g}\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{0}}}})$ and $\gamma _{\left (\theta _{1.8},\theta _{2.2}\right ]} \otimes \gamma _{\left (\theta _{2.8},\theta _{3.2}\right ]}$ commutes with $\left ( \gamma _{\left [0,\theta _1\right ]}\right )^{-1}\eta _{g}\gamma _{[0,\theta _1]}\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{1}}}})$, we have

(4.20)$$ \begin{align} \gamma^{-1}&\circ\eta_{g}\circ\gamma\notag\\ &=\text{equation }(4.19) = ({\textrm{{inner}}})\circ\gamma_{0}^{-1}\circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )^{-1}\notag\\ &\quad\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}\circ\eta_{g} \circ\left ( \gamma_{\left[0,\theta_1\right]} \right ) \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\otimes \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ) \gamma_{0}\notag\\ &=({\textrm{{inner}}})\circ\gamma_{0}^{-1}\circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right )^{-1}\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}\circ\eta_{g} \circ\left ( \gamma_{\left[0,\theta_1\right]} \right ) \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right ) \gamma_{0}. \end{align} $$

On the other hand, because $\gamma _{CS}$ and $\beta _{g}^{U}$ commute, we have

(4.21)$$ \begin{align} \gamma^{-1}\circ\beta_{g}^{U}\circ\gamma=({\textrm{{inner}}})\gamma_{0}^{-1}\circ\gamma_{CS}^{-1}\beta_{g}^{U}\gamma_{CS}\gamma_{0} =({\textrm{{inner}}})\gamma_{0}^{-1}\circ\beta_{g}^{U}\gamma_{0}. \end{align} $$

Combining equations (4.20) and (4.21), we obtain

(4.22)$$ \begin{align} \gamma^{-1}\tilde \beta_{g}\gamma &=({\textrm{{inner}}})\circ \gamma_{0}^{-1} \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right )^{-1}\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}\circ\eta_{g} \circ\left ( \gamma_{\left[0,\theta_1\right]} \right ) \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right ) \gamma_{0} \circ \gamma_{0}^{-1}\circ\beta_{g}^{U}\gamma_{0}\notag\\ &=({\textrm{{inner}}})\circ \gamma_{0}^{-1} \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right )^{-1}\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}\circ\eta_{g}\beta_{g}^{U} \circ\left ( \gamma_{\left[0,\theta_1\right]} \right ) \circ \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right ) \circ\gamma_{0}\notag\\ &=({\textrm{{inner}}}) \circ\left (\hat\eta_{g}^L\otimes \hat\eta_{g}^R\right )\circ \beta_{g}^{U}. \end{align} $$

In the second equality, we used the fact that $ \gamma _{[0,\theta _1]} \gamma _{(\theta _{0.8}, \theta _{1.2}]} $ and $\beta _{g}^{U}$ commute. This completes the proof of the claim.

Step 3. We use the setting and notation of steps 1 and 2 (with $\theta _{0}$ chosen in step 2). By Lemma 2.1, there exists

(4.23)$$ \begin{align} ( (W_g), (u_\sigma(g,h)))\in \mathop{\mathrm{IP}}\nolimits\left ( \omega, \alpha, \theta_{2}, (\tilde\beta_g ), (\eta_{g}^\sigma ), (\alpha_L,\alpha_R,\Theta) \right ). \end{align} $$

Now we have

(4.24)$$ \begin{align} \begin{split} &\omega\circ\gamma\in \mathop{\mathcal{SL}}\nolimits,\qquad \alpha\circ\gamma\in \mathop{\mathrm{EAut}}\nolimits(\omega\circ\gamma),\qquad \left ( \gamma^{-1}\circ\tilde \beta_{g}\circ\gamma\right )\in \mathop{\mathrm{IG}}\nolimits(\omega\circ\gamma,\theta_{1.2}),\\ &\left(\hat\eta_{g}^{\sigma}\right)\in {\mathcal T}\left (\theta_{1.2}, \left ( \gamma^{-1}\tilde \beta_{g}\gamma\right )\right ),\qquad \left(\hat \alpha_L,\hat \alpha_R,\hat\Theta\right)\in{\mathcal D}_{\alpha\gamma}^{\theta_{1.2}}. \end{split} \end{align} $$

We claim

(4.25)$$ \begin{align} ( (W_g), (u_\sigma(g,h)))\in \mathop{\mathrm{IP}}\nolimits\left ( \omega\circ\gamma, \alpha\circ\gamma, \theta_{1.2}, \left(\gamma^{-1}\tilde\beta_g\gamma\right), \left(\hat \eta_{g}^\sigma\right), \left(\hat \alpha_L,\hat \alpha_R,\hat \Theta\right) \right ). \end{align} $$

This immediately implies $h(\omega )=h(\omega \circ \gamma )$. To prove the claim, we first see from formulas (4.10) and (4.11) that

(4.26)$$ \begin{align} &\left ( \hat \alpha_L\otimes\hat \alpha_R\right )\circ \hat\Theta\circ \gamma_{0}^{-1} \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right )^{-1}\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}\notag\\ &=\alpha_{0}\circ \left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \right ) \circ \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \circ\gamma_{0} \circ\Theta\circ \left ( \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )\notag\\ &\quad \circ \gamma_{0}^{-1} \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right )^{-1}\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}\notag\\ &=\alpha_{0}\circ \left ( \gamma_{\left[0,\theta_1\right]}\otimes\gamma_{\left(\theta_1,\theta_2\right]} \right ) \circ \Theta\circ \left ( \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1}, \end{align} $$

because $\gamma _{\left (\theta _{0.8}, \theta _{1.2}\right ]}\circ \gamma _{0}\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{1.2}}}})$ and $\Theta \circ ( \gamma _{(\theta _2,\theta _3 ]}\otimes \gamma _{\left (\theta _3,\frac \pi 2\right ]} ) \circ ( \gamma _{\left (\theta _{1.8},\theta _{2.2}\right ]} \otimes \gamma _{\left (\theta _{2.8},\theta _{3.2}\right ]} )\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{1.8}}}^{c}})$ commute. Furthermore, because $\gamma _{\left [0,\theta _1\right ]}$ and $\Theta \circ ( \gamma _{(\theta _2,\theta _3 ]}\otimes \gamma _{\left (\theta _3,\frac \pi 2\right ]} ) \circ \left ( \gamma _{\left (\theta _{1.8},\theta _{2.2}\right ]} \otimes \gamma _{\left (\theta _{2.8},\theta _{3.2}\right ]} \right ) \in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{1.8}}}^{c}})$ commute and $\gamma _{\left (\theta _1,\theta _2\right ]}$ and $\Theta \in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{2}}}^{c}} )$ commute, we have

(4.27)$$ \begin{align} \left ( \hat \alpha_L\otimes\hat \alpha_R\right )&\circ \hat\Theta\circ \gamma_{0}^{-1} \left ( \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]} \right )^{-1}\circ \left ( \gamma_{\left[0,\theta_1\right]} \right )^{-1} =\text{equation }(4.26) \notag\\ & =\alpha_{0}\circ \gamma_{\left(\theta_1,\theta_2\right]} \circ \Theta\circ \left ( \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right )\notag\\ &=\alpha_{0}\circ \Theta\circ\gamma_{\left(\theta_1,\theta_2\right]} \circ\left ( \gamma_{\left(\theta_2,\theta_3\right]}\otimes \gamma_{\left(\theta_3,\frac\pi 2\right]} \right ) \circ \left ( \gamma_{\left(\theta_{1.8},\theta_{2.2}\right]} \otimes \gamma_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ) =\alpha_{0}\circ \Theta \circ\hat\gamma. \end{align} $$

Here $\hat \gamma :=\gamma _{(\theta _1,\theta _2 ]} \circ ( \gamma _{ (\theta _2,\theta _3 ]}\otimes \gamma _{\left (\theta _3,\frac \pi 2\right ]} ) \circ \left ( \gamma _{\left (\theta _{1.8},\theta _{2.2}\right ]} \otimes \gamma _{\left (\theta _{2.8},\theta _{3.2}\right ]} \right )\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{1}}}^{c}} )$ commutes with $\beta _{g}^{U}$. Combining this and

(4.28)$$ \begin{align} \hat \eta_{g}\beta_{g}^{U}= \left ( \gamma_{\left[0,\theta_1\right]}\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\gamma_{0}\right )^{-1} \eta_{g} \beta_{g}^{ U} \gamma_{\left[0,\theta_1\right]}\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right]}\gamma_{0}, \end{align} $$

we obtain

(4.29)$$ \begin{align} \pi_{0}\circ \left ( \hat \alpha_L\otimes\hat \alpha_R\right )\circ \hat\Theta\circ\hat\eta_{g}\beta_{g}^{U} ( \hat\Theta )^{-1} \left ( \hat \alpha_L\otimes\hat \alpha_R\right )^{-1} =\pi_{0}\circ\alpha_{0}\circ \Theta \circ\hat\gamma\circ \eta_{g} \beta_{g}^{ U}\circ \hat\gamma^{-1}\circ\Theta^{-1}\circ \alpha_{0}^{-1}. \end{align} $$

Because $\hat \gamma $ commutes with $\beta _{g}^{U}$ and $ \eta _{g}\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{0}}}} )$ commutes with $\hat \gamma \in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{1}}}^{c}} )$, we have

(4.30)$$ \begin{align} &\pi_{0}\circ \left ( \hat \alpha_L\otimes\hat \alpha_R\right )\circ \hat\Theta\circ\hat\eta_{g}\beta_{g}^{U} ( \hat\Theta )^{-1} \left ( \hat \alpha_L\otimes\hat \alpha_R\right )^{-1} \notag \\&\quad =\text{equation }(4.29) =\pi_{0}\circ\alpha_{0}\circ \Theta \circ \eta_{g} \beta_{g}^{ U} \circ\Theta^{-1}\circ \alpha_{0}^{-1} =\mathop{\mathrm{Ad}}\nolimits( W_{g} )\circ\pi_{0}. \end{align} $$

Hence the condition for $W_{g}$ in formula (4.25) is checked. On the other hand, substituting formulas (4.11) and (4.16), we get

(4.31)$$ \begin{align} \pi_R&\circ\hat \alpha_R\circ\hat \eta_g^R\beta_g^{R U} \hat \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \hat \eta_{gh}^R\right )^{-1}\hat\alpha_{R}^{-1}\notag\\ &= \pi_{R}\circ\alpha_R\circ \left ( \gamma_{\left[0,\theta_1\right],R}\otimes\gamma_{\left(\theta_1,\theta_2\right],R} \right ) \circ \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],R} \circ\gamma_{H,R} \circ \left ( \gamma_{\left[0,\theta_1\right],R}\circ\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],R}\circ\gamma_{H,R}\right )^{-1}\notag\\ & \quad \eta_g^R\beta_g^{RU} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\circ \gamma_{\left[0,\theta_1\right],R}\circ\gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],R} \notag\\ &\quad\circ\gamma_{H,R} \circ \left ( \left ( \gamma_{\left[0,\theta_1\right],R}\otimes\gamma_{\left(\theta_1,\theta_2\right],R} \right ) \circ \gamma_{\left(\theta_{0.8}, \theta_{1.2}\right],R} \circ\gamma_{H,R} \right )^{-1}\alpha_R^{-1}\notag\\ &= \pi_{R}\circ\alpha_R\circ \gamma_{\left(\theta_1,\theta_2\right],R} \circ \eta_g^R\beta_g^{RU} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1}\circ \left ( \gamma_{\left(\theta_1,\theta_2\right],R} \right )^{-1}\circ\alpha_R^{-1}. \end{align} $$

Because $ \eta _g^R\beta _g^{RU} \eta _h^R(\beta _g^{R U})^{-1}( \eta _{gh}^R)^{-1}\in \mathop {\mathrm {Aut}}\nolimits ( {\mathcal A}_{{ C_{\theta _{0}}}})$ commutes with $\gamma _{\left (\theta _1,\theta _2\right ],R}$, we obtain

(4.32)$$ \begin{align} \pi_R&\circ\hat \alpha_R\circ\hat \eta_g^R\beta_g^{R U} \hat \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \hat \eta_{gh}^R\right )^{-1}\hat\alpha_{R}^{-1} =\text{equation }(4.31)\notag\\ &=\pi_{R}\circ\alpha_R\circ \eta_g^R\beta_g^{RU} \eta_h^R\left (\beta_g^{R U}\right )^{-1}\left ( \eta_{gh}^R\right )^{-1} \alpha_R^{-1} =\mathop{\mathrm{Ad}}\nolimits\left ( u_{R}(g,h)\right )\circ\pi_{R}. \end{align} $$

An analogous statement for $\sigma =L$ also holds. This completes the proof of formula (4.25). Hence the statement of the theorem is proven.

5 Proof of Theorem 1.5

In this section, we prove Theorem 1.5. The proof relies heavily on the machinery of quasilocal automorphisms developed in [BMNS, NSY, MO]. (A summary is given in Appendix D.) We use terminology and facts from Appendixes C and D freely. We introduce a set of F-functions with fast decay, ${\mathcal F}_a$, as Definition C.2. A crucial point for us is the following:

Theorem 5.1. Set $\Phi _0,\Phi _1\in {\mathcal P}_{UG}$ and let $\omega _{\Phi _0}, \omega _{\Phi _1}$ be their unique gapped ground states. Suppose that $\Phi _0\sim \Phi _1$ holds, via a path $\Phi : [0,1]\to {\mathcal P}_{UG}$. Then there exists some $\Psi \in \hat {\mathcal B}_F([0,1])$ with $\Psi _{1}\in \hat {\mathcal B}_{F}([0,1])$ for some $F\in {\mathcal F}_a$ of the form $F(r)=\frac {\exp \left ( {-r^{\theta }}\right )}{(1+r)^{4}}$ with a constant $0<\theta <1$, such that $\omega _{\Phi _{1}}=\omega _{\Phi _0}\circ \tau _{1,0}^{\Psi }$. If $\Phi _0,\Phi _1\in {\mathcal P}_{UG\beta }$ and $\Phi \sim _\beta \Phi _0$, we may take $\Psi $ to be $\beta $-invariant.

For the proof, see Appendix D.

From this and Theorems 3.1 and 4.1, in order to show Theorem 1.5 it suffices to show the following, which says that the automorphism $\tau _{1,0}^{\Psi }$ in Theorem 5.1 satisfies all the good factorisation properties which we assumed in previous sections:

Theorem 5.2. Let $F\in {\mathcal F}_a$ be an F-function of the form $F(r)=\frac {\exp \left ( {-r^{\theta }}\right )}{(1+r)^{4}}$ with a constant $0<\theta <1$. Let $\Psi \in \hat {\mathcal B}_{F}([0,1])$ be a path of interactions satisfying $\Psi _1\in \hat {\mathcal B}_F([0,1])$. Then we have $\tau _{1,0}^{\Psi }\in \mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$. Furthermore, if $\Psi $ is $\beta _{g}^{U}$-invariant – that is, $\beta _{g}^U\left ( \Psi (X;t)\right )=\Psi (X;t)$ for any $X\in {\mathfrak S}_{{\mathbb Z}^2}$, $t\in [0,1]$ and $g\in G$ – then we have $\tau _{1,0}^{\Psi }\in \mathop {\mathrm {GSQAut}}\nolimits ({\mathcal A})$.

Proof. Fix arbitrary

(5.1)$$ \begin{align} 0<\theta_{0.8}<\theta_1<\theta_{1.2}<\theta_{1.8}<\theta_2<\theta_{2.2}< \theta_{2.8}<\theta_3<\theta_{3.2}<\frac\pi 2. \end{align} $$

We show the existence of the decomposition

(5.2)$$ \begin{align} \begin{split} \tau_{1,0}^\Psi=&\mathop{\mathrm{Ad}}\nolimits(u)\circ \left ( \alpha_{\left(0,\theta_1\right]}\otimes \alpha_{\left(\theta_1,\theta_2\right]}\otimes \alpha_{\left(\theta_2,\theta_3\right]} \otimes \alpha_{\left(\theta_3,\frac\pi 2\right]} \right )\\ &\circ \left ( \alpha_{\left(\theta_{0.8},\theta_{1.2}\right]} \alpha_{\left(\theta_{1.8},\theta_{2.2}\right]}\otimes \alpha_{\left(\theta_{2.8},\theta_{3.2}\right]} \right ), \end{split} \end{align} $$

with $\alpha $s of the forms in formulas (2.13) and (2.14). We follow the strategy of [NO].

Step 1. Fix some $0<\theta '<\theta $ and set

(5.3)$$ \begin{align} \tilde F(r):=\frac{\exp\left ( {-r^{\theta'}}\right )}{(1+r)^{4}}. \end{align} $$

With a suitably chosen constant $c_{1}>0$, we have

(5.4)$$ \begin{align} \max\left\{ F\left (\frac r 3\right ), \left ( F\left ( \left [ \frac r 3 \right ] \right )\right )^{\frac 12}\right\}\le c_{1}\tilde F(r),\quad r\ge 0. \end{align} $$

Namely, $c_{1}\tilde F(r)$ satisfy the condition on $\tilde F_{\theta }$ in Definition C.2(ii) for our $F=\frac {\exp \left ( {-cr^{\theta }}\right )}{(1+r)^{4}}$ and $\theta =\frac 12$. Set

(5.5)$$ \begin{align} {\mathcal C}_{0}&:=\left\{ \begin{gathered} C_{\left[0,\theta_1\right],\sigma}, \ C_{\left(\theta_1,\theta_2\right], \sigma,\zeta}, \ C_{\left(\theta_2,\theta_3\right], \sigma,\zeta}, \ C_{\left(\theta_3,\frac\pi 2\right], \zeta},\\ \sigma=L,R,\ \zeta=D,U \end{gathered} \right\}, \end{align} $$

(5.6)$$ \begin{align} {\mathcal C}_{1}&:=\left\{ \begin{gathered} C_{\left(\theta_{0.8},\theta_{1.2}\right),\sigma,\zeta}, \ C_{\left(\theta_{1.8},\theta_{2.2}\right), \sigma,\zeta}, \ C_{\left(\theta_{2.8},\theta_{3.2}\right), \sigma,\zeta},\\ \sigma=L,R,\ \zeta=D,U \end{gathered} \right\}. \end{align} $$

Define $\Psi ^{(0)}, \Psi ^{(1)}\in \hat {\mathcal B}_{F}([0,1])$ by

(5.7)$$ \begin{align} \begin{split} \Psi^{(0)}\left ( X; t\right )&:= \begin{cases} \Psi\left ( X; t\right ) & \text{if there exists a } C\in{\mathcal C}_0 \text{ such that } X\subset C, \\ 0 & \text{otherwise}, \end{cases} \\ \Psi^{(1)}\left ( X; t\right )&:=\Psi^{(0)}\left ( X; t\right )-\Psi\left ( X; t\right ), \end{split} \end{align} $$

for each $X\in {\mathfrak S}_{{\mathbb Z}^2}$, $t\in [0,1]$.

First we would like to represent $ ( \tau _{1,0}^{\Psi ^{(0)}})^{-1}\circ \tau _{1,0}^{ \Psi }$ as some quasilocal automorphism. Set $t,s\in [0,1]$. We apply Proposition D.6 for $\Psi $ replaced by $\Psi ^{(1)}$ and $\tilde \Psi $ by $\Psi $. Hence we set

(5.8)$$ \begin{align} \Xi^{(s)}\left ( Z, t \right ):= \sum_{m\ge 0} \sum_{X\subset Z,\, X(m)=Z} \Delta_{X(m)}\left ( \tau_{t,s}^{\Psi}\left ( \Psi^{(1)}\left ( X; t\right ) \right ) \right ) \end{align} $$

and

(5.9)$$ \begin{align} \Xi^{(n)(s)}\left ( Z, t \right ):= \sum_{m\ge 0} \sum_{X\subset Z, \, X(m)\cap\Lambda_{n}=Z} \Delta_{X(m)}\left ( \tau_{t,s}^{\left(\Lambda_n\right)\Psi}\left ( \Psi^{{(1)}}\left ( X; t\right ) \right ) \right ). \end{align} $$

Corresponding to equation (D.31), we obtain

(5.10)$$ \begin{align} \tau_{t,s}^{\left(\Lambda_n\right),\Psi} \left ( H_{\Lambda_n,\Psi^{(1)}}(t) \right ) = H_{\Lambda_n,\Xi^{(n)(s)}}(t). \end{align} $$

Applying Proposition D.6. we have $\Xi ^{(n)(s)}, \Xi ^{(s)}\in \hat {\mathcal B}_{\tilde F}([0,1])$, and

(5.11)$$ \begin{align} \lim_{n\to\infty}\left \lVert \tau_{t,u}^{\Xi^{(n)(s)}}\left ( A\right ) -\tau_{t,u}^{\Xi^{(s)}}\left ( A\right ) \right \rVert=0,\quad A\in{\mathcal A},\ t,u\in [0,1], \end{align} $$

holds. Two functions $\hat \tau _{t,s}^{(\Lambda _n), \Xi ^{(n)(s)}}(A)$ and $ \tau _{t,s}^{(\Lambda _n ), \Psi }\circ ( \tau _{t,s}^{(\Lambda _n ),\Psi ^{(0)}} )^{-1}(A)$ satisfy the same differential equation and initial condition. Therefore we obtain

(5.12)$$ \begin{align} \hat\tau_{t,s}^{\left(\Lambda_n\right), \Xi^{(n)(s)}}(A)= \tau_{t,s}^{\left(\Lambda_n\right), \Psi}\circ \left ( \tau_{t,s}^{\left(\Lambda_n\right),\Psi^{(0)}}\right )^{-1}(A),\quad t\in [0,1],\ A\in{\mathcal A}. \end{align} $$

From the fact that $ \hat \tau _{t,u}^{\left (\Lambda _n\right ), \Xi ^{(n)(s)}}=\tau _{u,t}^{\left (\Lambda _n\right ), \Xi ^{(n)(s)}}=\tau _{u,t}^{\Xi ^{(n)(s)}}$ converges strongly to an automorphism $\tau _{u,t}^{\Xi ^{(s)}}$ on ${\mathcal A}$ (equation (5.11)), we have

(5.13)$$ \begin{align} \lim_{n\to\infty}\left \lVert \hat\tau_{t,s}^{\left(\Lambda_{n}\right)\Xi^{(n)(s)}}\left ( A\right ) -\tau_{s,t}^{\Xi^{(s)}}\left ( A\right ) \right \rVert=0,\quad A\in{\mathcal A}. \end{align} $$

On the other hand, by Theorem D.3 we have, for $t \in [0,1]$ and $A \in {\mathcal A}$,

(5.14)$$ \begin{align} \lim_{n\to\infty}\left \lVert \tau_{t,s}^{\left(\Lambda_n\right), \Psi}\circ \left ( \tau_{t,s}^{\left(\Lambda_n\right),\Psi^{(0)}}\right )^{-1}(A) -\tau_{t,s}^{ \Psi}\circ \left ( \tau_{t,s}^{\Psi^{(0)}}\right )^{-1}(A) \right \rVert=0. \end{align} $$

Therefore, taking the $n\to \infty $ limit in equation (5.12), we obtain

(5.15)$$ \begin{align} \tau_{s,t}^{ \Xi^{(s)}}(A)= \tau_{t,s}^{\Psi}\circ \left ( \tau_{t,s}^{\Psi^{(0)}}\right )^{-1}(A),\quad t,s\in [0,1],\ A\in{\mathcal A}. \end{align} $$

Hence we have

(5.16)$$ \begin{align} \tau_{s,t}^{\Psi}=\left ( \tau_{t,s}^{\Psi}\right )^{-1} =\left ( \tau_{t,s}^{\Psi^{(0)}}\right )^{-1}\left (\tau_{s,t}^{ \Xi^{(s)}}\right )^{-1}= \tau_{s,t}^{\Psi^{(0)}}\tau_{t,s}^{ \Xi^{(s)}}. \end{align} $$

In particular, we get

(5.17)$$ \begin{align} \tau_{1,0}^{\Psi}= \tau_{1,0}^{\Psi^{(0)}}\tau_{0,1}^{ \Xi^{(1)}}. \end{align} $$

Step 2. We show

(5.18)$$ \begin{align} \sum_{\substack{Z\in{\mathfrak S}\left({\mathbb Z}^2\right), \\\not\exists C\in{\mathcal C}_{1}\text{ s.t. }Z\subset C}} \sup_{t\in \left[0,1\right]}\left \lVert \Xi^{(1)}\left ( Z,t\right )\right \rVert<\infty. \end{align} $$

From this,

(5.19)$$ \begin{align} V(t):=\sum_{\substack{Z\in{\mathfrak S}\left({\mathbb Z}^2\right), \\\not\exists C\in{\mathcal C}_{1}\text{ s.t. }Z\subset C}} \Xi^{(1)}\left ( Z,t\right ) \in{\mathcal A} \end{align} $$

converges absolutely in the norm topology and defines an element in ${\mathcal A}$. Furthermore, for

(5.20)$$ \begin{align} V_{n}(t):=\sum_{\substack{Z\in{\mathfrak S}\left({\mathbb Z}^2\right),\, Z\subset \Lambda_{n} \\\not\exists C\in{\mathcal C}_{1}\text{ s.t. }Z\subset C} } \Xi^{(1)}\left ( Z,t\right ) \in{\mathcal A}_{\Lambda_{n}},\quad n\in\mathbb{N}, \end{align} $$

we get

(5.21)$$ \begin{align} \lim_{n\to\infty}\sup_{t\in\left[0,1\right]}\left \lVert V_{n}(t)-V(t)\right \rVert=0 \end{align} $$

from formula (5.18).

To prove formula (5.18), we first bound

(5.22)$$ \begin{align} \begin{split} \sum_{\substack{Z\in{\mathfrak S}\left({\mathbb Z}^2\right) \\ \not\exists C\in{\mathcal C}_{1}\text{ s.t. }Z\subset C } } &\sup_{t\in\left[0,1\right]} \left \lVert \Xi^{(1)}\left ( Z,t\right ) \right \rVert \\ &\le \sum_{\substack{Z\in{\mathfrak S}\left({\mathbb Z}^2\right), \\ \not\exists C\in{\mathcal C}_{1}\text{ s.t. }Z\subset C } } \sum_{m\ge 0} \sum_{\substack{X: X\subset Z\\ X(m)=Z}} \left[ \sup_{t\in\left[0,1\right]}\left \lVert \Delta_{X(m)}\left ( \tau_{t,1}^{\Psi}\left ( \Psi^{(1)}\left ( X; t\right ) \right ) \right ) \right \rVert \right]\\ &\le \sum_{m\ge 0} \sum_{\substack{X: \\ \not\exists C\in{\mathcal C}_{1}\text{ s.t. }X(m)\subset C}} \sup_{t\in\left[0,1\right]}\left \lVert \Delta_{X(m)}\left ( \tau_{t,1}^{\Psi}\left ( \Psi^{(1)}\left ( X; t\right ) \right ) \right ) \right \rVert\\ & \le \sum_{m\ge 0} \sum_{\substack{X: \\ \not\exists C\in{\mathcal C}_{1}\text{ s.t. }X(m)\subset C}} \left[ \sup_{t\in\left[0,1\right]} \frac{8\left \lVert \Psi^{(1)}\left ( X; t\right ) \right \rVert}{C_{F}}\left ( e^{2I_{F}(\Psi)}-1\right )\left \lvert X\right \rvert G_{F}\left ( m\right ) \right] \\ &=\frac{8}{C_{F}}\left ( e^{2I_F(\Psi)}-1\right ) \sum_{m\ge 0} \sum_{\substack{X: \\ \not\exists C\in{\mathcal C}_{1}\text{ s.t. }X(m)\subset C}} \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi^{(1)}\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right]. \end{split} \end{align} $$

For the third inequality, we used Theorem D.3 3. For any cone $C_1,C_2$ of ${\mathbb Z}^2$ with its apex at the origin, we set

(5.23)$$ \begin{align} M(C_1,C_2):= \sum_{m\ge 0} \sum_{\substack{X: \\ \forall C\in{\mathcal C}_{1}, \, X\cap \left ( \left ( C^{c}\right )(m)\right )\neq\emptyset,\\ X\cap C_{1}\neq\emptyset,\, X\cap C_{2}\neq\emptyset }} \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi^{(1)}\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right]. \end{align} $$

From the definition of $ \Psi ^{(1)}$, we have $ \Psi ^{(1)}\left ( X; t\right ) =0, $ unless X has a nonempty intersection with at least two elements in ${\mathcal C}_{0}$. Therefore, if X gives a nonzero contribution in formula (5.22), then it has to satisfy

$$ \begin{gather*} X\cap \left ( \left ( C^{c}\right )(m)\right )\neq\emptyset,\quad \text{for all } C\in{\mathcal C}_{1},\\ \exists C_{1},C_{2}\in{\mathcal C}_{0}\text{ such that } C_1\neq C_2, \ X\cap C_{1}\neq\emptyset,\ X\cap C_{2}\neq\emptyset. \end{gather*} $$

Hence we have

(5.24)$$ \begin{align} \begin{split} \text{formula }(5.22)\le \frac{8}{C_{F}}\left ( e^{2I_F(\Psi)}-1\right ) \sum_{\substack{C_{1},C_{2}\in{\mathcal C}_{0}\\C_1\neq C_2 } } M({C_1,C_2}). \end{split} \end{align} $$

Hence it suffices to show that $M({C_1,C_2})<\infty $ for all $C_{1},C_{2}\in {\mathcal C}_{0}$ with $C_1\neq C_2$.

In order to proceed, we prepare two estimates. We will freely identify ${\mathbb C}$ and ${\mathbb R}^{2}$ in an obvious manner. In particular, $\arg z$ of $z\in {\mathbb Z}^{2}\subset {\mathbb R}^{2}$ in the following definition is considered with this identification: For $\varphi _{1}<\varphi _{2}$, we set

(5.25)$$ \begin{align} \check C_{\left[\varphi_{1},\varphi_{2}\right]}:= \left\{ z\in {\mathbb Z}^{2}\mid \arg z\in [\varphi_{1},\varphi_{2}] \right\}. \end{align} $$

We define $\check C_{\left (\varphi _{1},\varphi _{2}\right )}$ and so on analogously. Set

(5.26)$$ \begin{align} {c^{(0)}}_{\zeta_1,\zeta_2,\zeta_3,\zeta_4} :=\sqrt{1-\max\left\{\cos (\zeta_3-\zeta_2),\cos (\zeta_4-\zeta_1),0\right\}},\quad \zeta_1,\zeta_2,\zeta_3,\zeta_4 \in {\mathbb R}. \end{align} $$

Lemma 5.3. Set $\varphi _1<\varphi _2<\varphi _3<\varphi _4$ with $\varphi _4-\varphi _1<2\pi $. Then

$$ \begin{align*} \begin{split} b_0(\varphi_1,\varphi_2,\varphi_3,\varphi_4) &:= \sum_{m\ge 0} \sum_{\substack{X: \\ X\cap \check C_{\left[\varphi_1,\varphi_2\right]}\neq\emptyset\\ X\cap \check C_{\left[\varphi_3,\varphi_4\right]}\neq\emptyset }} \left[ \sup_{t\in\left[0,1\right]}\left (\left \lVert \Psi\left ( X; t\right )\right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right] \notag\\ &\le (64)^3 \frac {3^{4}\kappa_{1, 4, F}}{\left ( c^{(0)}_{\varphi_1,\varphi_2,\varphi_3,\varphi_4}\right ) ^{4}} \left ( \left \lVert\left \lvert \Psi_1\right \rVert\right \rvert_{F}\right ) \left ( \sum_{m\ge 0}G_{F}\left ( m\right )\right )<\infty. \end{split} \end{align*} $$

Proof. Substituting Lemma C.4, we obtain

(5.27)$$ \begin{align} b_0(\varphi_1,\varphi_2,\varphi_3,\varphi_4) &:= \sum_{m\ge 0} \sum_{\substack{X: \\ X\cap \check {C}_{\left[\varphi_1,\varphi_2\right]}\neq\emptyset\\ X\cap \check {C}_{\left[\varphi_3,\varphi_4\right]}\neq\emptyset }} \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right] \notag\\ &\le \sum_{m\ge 0} \sum_{\substack{x\in \check {C}_{\left[\varphi_1,\varphi_2\right]} \\y\in \check {C}_{\left[\varphi_3,\varphi_4\right]}}} \sum_{X\ni x,y}\left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right]\notag\\ &\le \left ( \left \lVert\left \lvert \Psi_1\right \rVert\right \rvert_{F}\right ) \sum_{\substack{x\in \check {C}_{\left[\varphi_1,\varphi_2\right]}\\ y\in \check {C}_{\left[\varphi_3,\varphi_4\right]}}} F\left ( {\mathrm{d}}(x,y)\right ) \left ( \sum_{m\ge 0}G_{F}\left ( m\right )\right )\notag\\ &\le (64)^3 \frac {3^{4}\kappa_{1, 4, F}}{\left ( c^{(0)}_{\varphi_1,\varphi_2,\varphi_3,\varphi_4}\right ) ^{4}} \left ( \left \lVert\left \lvert \Psi_1\right \rVert\right \rvert_{F}\right ) \left ( \sum_{m\ge 0}G_{F}\left ( m\right )\right )<\infty. \end{align} $$

We used Lemma C.4 in the last inequality. The last value is finite by equation (C.14) for our $F\in {\mathcal F}_{a}$.

Set

(5.28)$$ \begin{align} {c^{(1)}}_{\zeta_1,\zeta_2,\zeta_3} :=\sqrt{1-\max\left\{\cos (\zeta_1-\zeta_2),\cos (\zeta_1-\zeta_3)\right\}},\quad \zeta_1,\zeta_2,\zeta_3\in [0,2\pi). \end{align} $$

Lemma 5.4. For $\varphi _1<\varphi _2<\varphi _3$ with $\varphi _3-\varphi _1<\frac \pi 2$, we have

(5.29)$$ \begin{align} \begin{split} b_1(\varphi_1,\varphi_2,\varphi_3) &:=\sum_{m\ge 0} \sum_{\substack{X: \\ X\subset \check {C}_{\left[\varphi_1,\varphi_3\right]}\\ X\cap \check {C}_{\left[\varphi_1,\varphi_2\right]}\neq\emptyset\\ X\cap \check C_{\left[\varphi_2,\varphi_3\right]}\neq\emptyset\\ X\cap \left ( \left ( \left ( \check C_{\left(\varphi_1,\varphi_3\right)}\right )^c\right )(m)\right )\neq \emptyset }} \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right]\\ &\le64\cdot 144\cdot 24\cdot \left ( \pi \kappa_{1,2,F}+F(0) \right ) \left ( \left \lVert\left \lvert \Psi_1\right \rVert\right \rvert_F\right ) \left ( \sum_{m\ge 0} (m+1)^{4}G_{F}\left ( m\right )\right ) \\ &\quad \left ( \left ( {c^{(1)}}_{\varphi_{1},\varphi_{2},\varphi_{3}}\right )^{-4} +\left ( {c^{(1)}}_{\varphi_{3},\varphi_{1},\varphi_2}\right )^{-4} \right ) <\infty. \end{split} \end{align} $$

Proof. Set

(5.30)$$ \begin{align} L_\varphi:= \left\{ z\in{\mathbb R}^2\mid \arg z=\varphi \right\},\quad \varphi\in [0,2\pi). \end{align} $$

Note that if $X\in {\mathfrak S}_{{\mathbb Z}^2}$ satisfies $X\subset \check {C}_{\left [\varphi _1,\varphi _3\right ]}$ and $X\cap ( ( ( \check C_{ (\varphi _1,\varphi _3 )} )^c )(m) )\neq \emptyset $, then we have

(5.31)$$ \begin{align} d(X, L_{\varphi_1})\le m\quad\text{or}\quad d(X, L_{\varphi_3})\le m. \end{align} $$

Therefore, we have

(5.32)$$ \begin{align}\sum_{m\ge 0}& \sum_{\substack{X: \\ X\subset \check {C}_{\left[\varphi_1,\varphi_3\right]}\\ X\cap \check {C}_{\left[\varphi_1,\varphi_2\right]}\neq\emptyset\\ X\cap \check C_{\left[\varphi_2,\varphi_3\right]}\neq\emptyset\\ X\cap \left ( \left ( \left ( \check C_{\left(\varphi_1,\varphi_3\right)}\right )^c\right )(m)\right )\neq \emptyset }} \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right] \nonumber \\ &\le\sum_{m\ge 0} G_{F}\left ( m\right ) \left ( \sum_{\substack{X: \\ X\cap \check C_{\left[\varphi_2,\varphi_3\right]}\neq\emptyset\\ d\left(X, L_{\varphi_1}\right)\le m }} +\sum_{\substack{X: \\ X\cap \check{C}_{\left[\varphi_1,\varphi_2\right]}\neq\emptyset\\ d\left(X, L_{\varphi_3}\right)\le m }} \right ) \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert \right] \nonumber\\ &\le \sum_{m\ge 0} G_{F}\left ( m\right ) \left ( \sum_{\substack{x\in \check C_{\left[\varphi_2,\varphi_3\right]}\\ y\in L_{\varphi_1}(m) }} +\sum_{\substack{x\in \check {C}_{\left[\varphi_1,\varphi_2\right]}\\ y\in L_{\varphi_3}(m) }} \right ) \sum_{\substack{X: X\ni x,y }} \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert \right] \nonumber\\ &\le \left ( \left \lVert\left \lvert \Psi_1\right \rVert\right \rvert_F\right ) \sum_{m\ge 0} G_{F}\left ( m\right ) \left ( \sum_{\substack{x\in \check C_{\left[\varphi_2,\varphi_3\right]}\\ y\in L_{\varphi_1}(m) }} +\sum_{\substack{x\in \check {C}_{\left[\varphi_1,\varphi_2\right]}\\ y\in L_{\varphi_3}(m) }} \right ) F\left ( {\mathrm{d}}(x,y)\right )\nonumber\\ &\le 64\cdot 144\cdot 24\cdot \left ( \pi \kappa_{1,2,F}+F(0) \right ) \left ( \left \lVert\left \lvert \Psi_1\right \rVert\right \rvert_F\right )\nonumber \\ &\quad \left ( \sum_{m\ge 0} (m+1)^{4}G_{F}\left ( m\right )\right ) \left ( \left ( {c^{(1)}}_{\varphi_{1},\varphi_{2},\varphi_{3}}\right )^{-4} +\left ( {c^{(1)}}_{\varphi_{3},\varphi_{1},\varphi_2}\right )^{-4} \right ). \end{align} $$

In the last inequality, we used Lemma C.5 with $\varphi _3-\varphi _1<\frac \pi 2$. Because $\varphi _3-\varphi _1<\frac \pi 2$ and because of formula (C.14), the last value is finite.

Now let us go back to the estimate of formula (5.23). If $C_1,C_2\in {\mathcal C}_0$ are $C_1=\check {C}_{\left [\varphi _1,\varphi _2\right ]}, C_2=\check {C}_{\left [\varphi _3,\varphi _4\right ]}$ with $\varphi _1<\varphi _2<\varphi _3<\varphi _4, \varphi _4-\varphi _1<2\pi $, then from Lemma 5.3, we have

(5.33)$$ \begin{align} \begin{split} & M(C_1,C_2) \le b_0(\varphi_1,\varphi_2,\varphi_3,\varphi_4) <\infty. \end{split} \end{align} $$

Now suppose that $C_1,C_2\in {\mathcal C}_0$ are $C_1=\check {C}_{\left [\varphi _1,\varphi _2\right ]}, C_2=\check C_{\left [\varphi _2,\varphi _3\right ]}$ with $\varphi _1<\varphi _2<\varphi _3, \varphi _3-\varphi _1<2\pi $. (Recall definition (5.5).) By the definition of ${\mathcal C}_0$ and ${\mathcal C}_1$, there is some $C=C_{\left (\zeta _1,\zeta _2\right )}\in {\mathcal C}_1$ such that $\varphi _1<\zeta _1<\varphi _2<\zeta _2<\varphi _3$ and $\zeta _2-\zeta _1<\frac \pi 2$. For $X\in {\mathfrak S}_{{\mathbb Z}^2}$ to give a nonzero contribution in formula (5.23), it has to satisfy

(5.34)$$ \begin{align} X(m)\cap ( \check C_{\left[\zeta_1,\zeta_2\right]})^c\neq\emptyset,\qquad X\cap \check {C}_{\left[\varphi_1,\varphi_2\right]}\neq\emptyset,\qquad X\cap \check C_{\left[\varphi_2,\varphi_3\right]}\neq\emptyset. \end{align} $$

For such an X, one of the following occurs:

1. (i) $X\cap \check C_{\left [\zeta _2,\varphi _3\right ]}\neq \emptyset $ and $X\cap \check {C}_{\left [\varphi _1,\varphi _2\right ]}\neq \emptyset $.

2. (ii) $X\cap \check C_{\left [\varphi _1,\zeta _1\right ]}\neq \emptyset $ and $X\cap \check C_{\left [\varphi _2,\varphi _3\right ]}\neq \emptyset $.

3. (iii) $X\cap \check C_{\left [\varphi _2, \zeta _2\right ]}\neq \emptyset (\text {and } X\cap \check C_{\left [\zeta _1,\varphi _2\right ]}\neq \emptyset )$ and $X\cap \check C_{\left [\varphi _3,\varphi _1+2\pi \right ]}\neq \emptyset $.

4. (iv) $X\subset \check C_{\zeta _1,\zeta _2}$, $X\cap ( ( \check C_{\zeta _1,\zeta _2} )^c )(m)\neq \emptyset $, $X\cap \check C_{\left [\varphi _2, \zeta _2\right ]}\neq \emptyset $ and $X\cap \check C_{\left [\zeta _1,\varphi _2\right ]}\neq \emptyset $.

Hence we get

(5.35)$$ \begin{align} \begin{split} M({C_1,C_2}) &\le b_0(\varphi_1,\varphi_2, \zeta_2,\varphi_3) +b_0(\varphi_1,\zeta_1, \varphi_2,\varphi_3) +b_0(\varphi_2, \zeta_2, \varphi_3,\varphi_1+2\pi) +b_1(\zeta_1,\varphi_2,\zeta_2)\\ &<\infty. \end{split} \end{align} $$

Hence we have proven the claim of step 2.

Step 3. Next we set

(5.36)$$ \begin{align} \tilde\Xi(Z,t):= \begin{cases} \Xi^{(1)}(Z,t) & \text{if }\exists C\in{\mathcal C}_{1}\text{ s.t. } Z\subset C,\\ 0&\text{otherwise}. \end{cases} \end{align} $$

Clearly, we have $\tilde \Xi \in \hat {\mathcal B}_{\tilde F}([0,1])$. Note that

(5.37)$$ \begin{align} H_{\Lambda_{n}, \tilde\Xi}(t)+V_{n}(t) = H_{\Lambda_{n}, \Xi^{(1)}}(t). \end{align} $$

As a uniform limit of $[0,1]\ni t\mapsto V_{n}(t)\in {\mathcal A}$ (equation (5.21)), $[0,1]\ni t\mapsto V(t)\in {\mathcal A}$ is norm-continuous. Because $\tilde \Xi \in \hat {\mathcal B}_{\tilde F}([0,1])$, $[0,1]\ni t\mapsto \tau _{t,s}^{\tilde \Xi }\left ( V(t)\right )\in {\mathcal A}$ is also norm-continuous, for each $s\in [0,1]$. Therefore, for each $s\in [0,1]$, there is a unique norm-differentiable map $[0,1]\ni t \mapsto W^{(s)}(t) \in {\mathcal U}\left ( {\mathcal A}\right )$ such that

(5.38)$$ \begin{align} \frac{d}{dt} W^{(s)}(t)=-i \tau_{t,s}^{\tilde\Xi}\left ( V(t)\right ) W^{(s)}(t),\quad W^{(s)}(s)=\mathbb I. \end{align} $$

It is given by

(5.39)$$ \begin{align} W^{(s)}(t) :=\sum_{k=0}^{\infty }(-i)^{k} \int_{s}^{t}ds_{1}\int_{s}^{s_{1}}ds_{2}\dotsi \int_{s}^{s_{k-1}}ds_{k} \tau_{s_{1},s}^{\tilde\Xi}\left ( V(s_{1})\right ) \dotsm \tau_{s_{k},s}^{\tilde\Xi}\left ( V(s_{k})\right ). \end{align} $$

Analogously, for each $s\in [0,1]$ and $n\in \mathbb {N}$, we define a unique norm-differentiable map from $[0,1]$ to $ {\mathcal U}\left ( {\mathcal A}\right )$ such that

(5.40)$$ \begin{align} \frac{d}{dt} W_{n}^{(s)}(t)=-i \tau_{t,s}^{(\Lambda_{n})\tilde\Xi}\left ( V_{n}(t)\right ) W_{n}^{(s)}(t),\quad W_{n}^{(s)}(s)=\mathbb I. \end{align} $$

It is given by

(5.41)$$ \begin{align} W_{n}^{(s)}(t) :=\sum_{k=0}^{\infty }(-i)^{k} \int_{s}^{t}ds_{1}\int_{s}^{s_{1}}ds_{2}\dotsi \int_{s}^{s_{k-1}}ds_{k} \tau_{s_{1},s}^{\left(\Lambda_{n}\right)\tilde\Xi}\left ( V_{n}(s_{1})\right ) \dotsm \tau_{s_{k},s}^{\left(\Lambda_{n}\right)\tilde\Xi}\left ( V_{n}(s_{k})\right ). \end{align} $$

By the uniform convergence (5.21) and Lemma D.3, we have

(5.42)$$ \begin{align}\lim_{n\to\infty} \sup_{t\in\left[0,1\right]}\left \lVert \tau_{t,s}^{\left(\Lambda_{n}\right)\tilde\Xi}\left ( V_{n}(t)\right ) -\tau_{t,s}^{\tilde\Xi}\left ( V(t)\right )\right \rVert=0. \end{align} $$

From this and formulas (5.39) and (5.41), we obtain

(5.43)$$ \begin{align} \lim_{n\to\infty}\sup_{t\in\left[0,1\right]}\left \lVert W_{n}^{(s)}(t)- W^{(s)}(t) \right \rVert =0. \end{align} $$

This and Theorem D.3 4 for $\Xi ^{(1)}, \tilde \Xi \in {\mathcal B}_{\tilde F}([0,1])$ imply

(5.44)$$ \begin{align} \begin{split} \lim_{n\to\infty} \tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A)&= \tau_{s,t}^{ \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W^{(s)}(t)\right ) (A),\\ \lim_{n\to\infty} \tau_{s,t}^{\left(\Lambda_{n}\right), \Xi^{(1)}}(A)&= \tau_{s,t}^{ \Xi^{(1)}}(A), \end{split} \end{align} $$

for any $A\in {\mathcal A}$.

Note that for any $A\in {\mathcal A}$,

$$ \begin{align*} \begin{split} \frac{d}{dt} \tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A) &=-i\left[ H_{\Lambda_{n}, \tilde\Xi}(t), \tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A) \right] \\ &\quad -i\tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\left ( \left[ \tau_{t,s}^{\left(\Lambda_{n}\right), \tilde\Xi}\left ( V_{n}(t)\right ), \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A) \right] \right ) \\ &=-i\left[ H_{\Lambda_{n}, \tilde\Xi}(t)+V_{n}(t), \tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A) \right]\nonumber\\ &=-i\left[ H_{\Lambda_{n}, \Xi^{(1)}}(t), \tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A) \right]. \end{split} \end{align*} $$

We used equation (D.10) for the second equality and equation (5.37) for the third. On the other hand, for any $A\in {\mathcal A}$, we have

(5.45)$$ \begin{align} \frac{d}{dt} \tau_{s,t}^{\left(\Lambda_{n}\right), \Xi^{(1)}} (A)=-i\left[ H_{\Lambda_{n}, \Xi^{(1)}}(t), \tau_{s,t}^{\left(\Lambda_{n}\right), \Xi^{(1)}} (A) \right]. \end{align} $$

Therefore, $\tau _{s,t}^{\left (\Lambda _{n}\right ), \tilde \Xi }\circ \mathop {\mathrm {Ad}}\nolimits ( W_{n}^{(s)}(t) ) (A)$ and $ \tau _{s,t}^{\left (\Lambda _{n}\right ), \Xi ^{(1)}} (A)$ satisfy the same differential equation. Also note that we have $\tau _{s,s}^{\left (\Lambda _{n}\right ), \tilde \Xi }\circ \mathop {\mathrm {Ad}}\nolimits ( W_{n}^{(s)}(s) ) (A)= \tau _{s,s}^{\left (\Lambda _{n}\right ), \Xi ^{(1)}} (A)=A$. Therefore, we get

(5.46)$$ \begin{align} \tau_{s,t}^{\left(\Lambda_{n}\right), \tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W_{n}^{(s)}(t)\right ) (A) =\tau_{s,t}^{\left(\Lambda_{n}\right), \Xi^{(1)}} (A). \end{align} $$

By equation (5.44), we obtain

(5.47)$$ \begin{align} \tau_{s,t}^{\tilde\Xi}\circ \mathop{\mathrm{Ad}}\nolimits \left ( W^{(s)}(t)\right ) (A) =\tau_{s,t}^{\Xi^{(1)}} (A),\quad A\in{\mathcal A}, \ t,s\in[0,1]. \end{align} $$

Taking the inverse, we get

(5.48)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits \left ( W^{(s)^{*}}(t)\right )\circ\tau_{t,s }^{\tilde\Xi} =\tau_{t,s}^{\Xi^{(1)}},\quad t,s\in[0,1]. \end{align} $$

Step 4. Combining equations (5.17) and (5.48), we have

(5.49)$$ \begin{align} \tau_{1,0}^{\Psi}= \tau_{1,0}^{\Psi^{(0)}}\tau_{0,1}^{ \Xi^{(1)}} =\tau_{1,0}^{\Psi^{(0)}}\circ\mathop{\mathrm{Ad}}\nolimits \left ( \left ( W^{(1)}(0)\right )^{*}\right )\circ\tau_{0,1 }^{\tilde\Xi}. \end{align} $$

By the definitions of $\Psi ^{(0)}$ and $\tilde \Xi $, we obtain decompositions

(5.50)$$ \begin{align} \begin{split} \tau_{1,0}^{\Psi^{(0)}} &=\alpha_{\left[0,\theta_{1}\right]}\otimes \alpha_{\left(\theta_1,\theta_2\right]}\otimes \alpha_{\left(\theta_2,\theta_3\right]} \otimes \alpha_{\left(\theta_3,\frac\pi 2\right]},\\ \tau_{0,1}^{\tilde\Xi} &= \alpha_{\left(\theta_{0.8},\theta_{1.2}\right]}\otimes \alpha_{\left(\theta_{1.8},\theta_{2.2}\right]}\otimes \alpha_{\left(\theta_{2.8},\theta_{3.2}\right]}, \end{split} \end{align} $$

with $\alpha $s in the form of formulas (2.13) and (2.14). This completes the proof of the first part.

Step 5. Suppose that $\beta _{g}^{U}\left ( \Psi (X;t)\right )=\Psi (X;t)$ for any $X\in {\mathfrak S}_{{\mathbb Z}^2}$, $t\in [0,1]$ and $g\in G$. Then clearly we have $\beta _{g}^{U}\left ( \Psi ^{(0)}(X;t)\right )=\Psi ^{(0)}(X;t)$ for any $X\in {\mathfrak S}_{{\mathbb Z}^2}$, $t\in [0,1]$ and $g\in G$. By Theorem D.3 5, this implies $\tau _{1,0}^{\Psi ^{(0)}}{\beta _g^U}={\beta _g^U}\tau _{1,0}^{\Psi ^{(0)}}$. From the decomposition (5.50), this means that all of $\alpha _{\left [0,\theta _{1}\right ],\sigma }, \alpha _{\left (\theta _1,\theta _2\right ],\sigma ,\zeta }, \alpha _{\left (\theta _2,\theta _3\right ],\sigma ,\zeta }, \alpha _{\left (\theta _3,\frac \pi 2\right ],\zeta }$, $\sigma =L,R, \zeta =U,D$, commute with ${\beta _g^U}$. Because $\Pi _{X}$ commutes with $\beta _{g}^{U}$, $\tau _{t,s}^{\Psi }$ commutes with $\beta _{g}^{U}$ (Theorem D.3 5), and $\Psi ^{(1)}$ and $\Xi ^{(s)}$ are $\beta _{g}^{U}$-invariant from the definition (5.8). Therefore, from the definition (5.36), $\tilde \Xi $ is also $\beta _{g}^{U}$-invariant. Hence by Theorem D.3 5, $\tau _{0,1}^{ \tilde \Xi }$ commutes with ${\beta _g^U}$. The decomposition (5.50) then implies that $\alpha _{\left (\theta _{0.8},\theta _{1.2}\right ],\sigma ,\zeta }, \alpha _{\left (\theta _{1.8},\theta _{2.2}\right ],\sigma ,\zeta }, \alpha _{\left (\theta _{2.8},\theta _{3.2}\right ],\sigma ,\zeta }$, $\sigma =L,R, \ \zeta =U,D$, commute with ${\beta _g^U}$.

An analogous proof shows the following:

Proposition 5.5. Let $F\in {\mathcal F}_a$ be an F-function of the form $F(r)=\frac {\exp \left ( {-r^{\theta }}\right )}{(1+r)^{4}}$ with a constant $0<\theta <1$. Let $\Psi \in \hat {\mathcal B}_{F}([0,1])$ be a path of interactions satisfying $\Psi _1\in \hat {\mathcal B}_F([0,1])$. Define $\Psi ^{(0)}\in \hat {\mathcal B}_{F}([0,1])$ by

(5.51)$$ \begin{align} \begin{split} & \Psi^{(0)}\left ( X; t\right ):= \begin{cases} \Psi\left ( X; t\right ) & \text{if } X\subset H_U\text{ or } X\subset H_D, \\ 0& \text{otherwise}, \end{cases} \end{split} \end{align} $$

for each $X\in {\mathfrak S}_{{\mathbb Z}^2}, t\in [0,1]$. Then $ \left (\tau _{1,0}^{\Psi ^{(0)}}\right )^{-1}\tau _{1,0}^{\Psi }$ belongs to $\mathop {\mathrm {HAut}}\nolimits ({\mathcal A})$.

Proof. Define $\tilde F$ as in formula (5.3) with some $0<\theta '<\theta $. The same argument as in Theorem 5.2, step 2, implies that there exists $\Xi ^{(1)}\in \hat {\mathcal B}_{\tilde F}[0,1]$ with $\tilde F\in {\mathcal F}_a$, such that

(5.52)$$ \begin{align} \tau_{1,0}^{\Psi}= \tau_{1,0}^{\Psi^{(0)}}\tau_{0,1}^{ \Xi^{(1)}}. \end{align} $$

This $\Xi ^{(1)}$ is given by formula (5.8) for current $\Psi $ and $\Psi ^{(1)}\left ( X; t\right ):=\Psi ^{(0)}\left ( X; t\right )-\Psi \left ( X; t\right )$. To prove the theorem, it suffices to show that $\tau _{0,1}^{ \Xi ^{(1)}}$ belongs to $\mathop {\mathrm {HAut}}\nolimits ({\mathcal A})$. Indeed, for any $0<\theta _0<\frac \pi 4$, as in Theorem 5.2, step 2, we have

(5.53)$$ \begin{align} &\sum_{\substack{Z: Z\nsubseteq C_{\left[0,\theta_0\right],L} \\Z\nsubseteq C_{\left[0,\theta_0\right],R} }} \sup_{t\in \left[0,1\right]}\left \lVert \Xi^{(1)}\left ( Z,t\right )\right \rVert\nonumber\\ &\qquad\qquad\le \frac{8}{C_{F}}\left ( e^{2I_F(\Psi)}-1\right ) \sum_{m\ge 0} \sum_{\substack{X: X(m)\nsubseteq C_{\left[0,\theta_0\right],L} \\X(m)\nsubseteq C_{\left[0,\theta_0\right],R} } } \left[ \sup_{t\in\left[0,1\right]}\left ( \left \lVert \Psi^{(1)}\left ( X; t\right ) \right \rVert\right ) \left \lvert X\right \rvert G_{F}\left ( m\right ) \right] <\infty. \end{align} $$

To see this, note that if X in the last line has a nonzero contribution to the sum, then at least one of the following occurs:

1. (i) $X\cap C_{\left [\theta _{0}, \frac \pi 2\right ],U}\neq \emptyset $ and $X\cap H_{D}\neq \emptyset $.

2. (ii) $X\cap C_{\left [\theta _{0}, \frac \pi 2\right ],D}\neq \emptyset $ and $X\cap H_{U}\neq \emptyset $.

3. (iii) $X\subset C_{\left [0,\theta _{0}\right ]}$ and

   1. (1) $X\cap C_{\left [0,\theta _{0}\right ],L}\neq \emptyset $ and $X\cap C_{\left [0,\theta _{0}\right ],R}\neq \emptyset $, or

   2. (2) $X\subset C_{\left [0,\theta _{0}\right ], R}$, $X\cap \check C_{\left [0,\theta _{0}\right ]}\neq \emptyset $, $X\cap \check C_{\left [-\theta _{0},0\right ]}\neq \emptyset $ and $X{(m)}\cap \left ( C_{\left [0,\theta _{0}\right ], R}\right )^{c}\neq \emptyset $, or

   3. (3) $X\subset C_{\left [0,\theta _{0}\right ], L}$, $X\cap \check C_{\left [\pi -\theta _{0},\pi \right ]}\neq \emptyset $, $X\cap \check C_{\left [\pi , \pi +\theta _{0}\right ]}\neq \emptyset $ and $X{(m)}\cap \left ( C_{\left [0,\theta _{0}\right ], L}\right )^{c}\neq \emptyset $.

Therefore, the summation in the second line of formula (5.53) is bounded by

$$ \begin{align*} \frac{8}{C_{F}}\left ( e^{2I_F(\Psi)}-1\right ) \left ( \begin{gathered} b_{0}(\theta_{0}, \pi-\theta_{0}, \pi, 2\pi) +b_{0}(0,\pi, \pi+\theta_{0}, 2\pi-\theta_{0}) +b_{0}(-\theta_{0}, \theta_{0}, \pi-\theta_{0}, \pi+\theta_{0})\\ +b_{1}(-\theta_{0}, 0,\theta_{0}) +b_{1}(\pi-\theta_{0}, \pi, \pi+\theta_{0}) \end{gathered} \right )<\infty, \end{align*} $$

from Lemmas 5.3 and 5.4, proving formula (5.53).

Therefore, as in step 3 of Theorem 5.2, setting

(5.54)$$ \begin{align} \tilde\Xi(Z,t):= \begin{cases} \Xi^{(1)}(Z,t)&\text{if } Z\subseteq C_{\left[0,\theta_0\right],L} \text{ or }Z\subseteq C_{\left[0,\theta_0\right],R}, \\ 0&\text{otherwise}, \end{cases} \end{align} $$

we obtain $\tau _{0,1}^{ \Xi ^{(1)}}=({\textrm {{inner}}})\circ \tau _{0,1}^{ \tilde \Xi }$. By the definition, $\tau _{0,1}^{ \tilde \Xi }$ decomposes as $\tau _{0,1}^{ \tilde \Xi }= \zeta _L\otimes \zeta _R$, with some $\zeta _\sigma \in \mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{C_{\left [0,\theta _0\right ],\sigma }} )$, $\sigma =L,R$. As this holds for any $0<\theta _0<\frac \pi 4$, we conclude $\tau _{0,1}^{ \Xi ^{(1)}}\in \mathop {\mathrm {HAut}}\nolimits ({\mathcal A})$.

Theorem 5.6. Let $F\in {\mathcal F}_a$ be an F-function of the form $F(r)=\frac {\exp \left ( {-r^{\theta }}\right )}{(1+r)^{4}}$ with a constant $0<\theta <1$. Let $\Psi \in \hat {\mathcal B}_{F}([0,1])$ be a path of interactions satisfying $\Psi _1\in \hat {\mathcal B}_F([0,1])$. If $\Psi $ is $\beta $-invariant, then $\tau _{1,0}^{\Psi }$ belongs to $\mathop {\mathrm {GUQAut}}\nolimits ({\mathcal A})$.

Proof. Define $\Psi ^{(0)}$ as in formula (5.51) for our $\Psi $. By Proposition 5.5, we have $ (\tau _{1,0}^{\Psi ^{(0)}} )^{-1}\tau _{1,0}^{\Psi }\in \mathop {\mathrm {HAut}}\nolimits ({\mathcal A})$. On the other hand, applying Theorem 5.2 to $\Psi ^{(0)}\in \hat {\mathcal B}_F([0,1])$, we see that $\tau _{1,0}^{\Psi ^{(0)}}$ belongs to $\mathop {\mathrm {SQAut}}\nolimits ({\mathcal A}).$ Note that $\Psi ^{(0)}(X;t)$ is nonzero only if $X\subset H_U$ or $X\subset H_D$, and it coincides with $\Psi (X;t)$ when it is nonzero. Therefore, if $\Psi $ is $\beta $-invariant, $\Psi ^{(0)}$ is $\beta _g^U$-invariant. Therefore, by Theorem 5.2, we have $\tau _{1,0}^{\Psi ^{(0)}}\in \mathop {\mathrm {GSQAut}}\nolimits ({\mathcal A})$. Hence we have $\tau _{1,0}^{\Psi }\in \mathop {\mathrm {GUQAut}}\nolimits ({\mathcal A})$.

Proof of Theorem 1.5.

Let $\Phi _0\in {\mathcal P}_{UG}$ be the fixed trivial interaction with a unique gapped ground state. Its ground state $\omega _0:=\omega _{\Phi _0}$ is of a product form (formula (2.18)). For any $\Phi \in {\mathcal P}_{SL\beta }$, we have $\Phi _0\sim \Phi $. Then by Theorem 5.1, there exists some $\Psi \in \hat {\mathcal B}_F([0,1])$ with $\Psi _{1}\in \hat {\mathcal B}_{F}([0,1])$ for some $F\in {\mathcal F}_a$ of the form $F(r)=\frac {\exp \left ( {-r^{\theta }}\right )}{(1+r)^{4}}$ with $0<\theta <1$, such that $\omega _{\Phi }=\omega _{\Phi _0}\circ \tau _{1,0}^{\Psi }$. From Theorem 5.2, $\tau _{1,0}^\Psi $ belongs to $\mathop {\mathrm {SQAut}}\nolimits ({\mathcal A})$. Because $\Phi \in {\mathcal P}_{SL\beta }$, $\omega _{\Phi }=\omega _{\Phi _0}\circ \tau _{1,0}^{\Psi }$ is $\beta $-invariant. Then, by Theorem 3.1, $\mathop {\mathrm {IG}}\nolimits (\omega _{\Phi })$ is not empty. Therefore, we may define $h_\Phi :=h(\omega _{\Phi })$ by Definition 2.18.

To see that $h_\Phi $ is an invariant of $\sim _\beta $, set $\Phi _1,\Phi _2\in {\mathcal P}_{SL\beta }$ with $\Phi _1\sim _\beta \Phi _2$. Then by Theorem 5.1, there exists some $\beta $-invariant $\Psi \in \hat {\mathcal B}_F([0,1])$ with $\Psi _{1}\in \hat {\mathcal B}_F([0,1])$ for some $F\in {\mathcal F}_a$ of the form $F(r)=\frac {\exp \left ( {-r^{\theta }}\right )}{(1+r)^{4}}$ with a constant $0<\theta <1$ such that $\omega _{\Phi _2}=\omega _{\Phi _1}\circ \tau _{1,0}^{\Psi }$. Applying Theorem 5.6 to this $\Psi $, we see that $\tau _{1,0}^{\Psi }$ belongs to $\mathop {\mathrm {GUQAut}}\nolimits ({\mathcal A})$. Then Theorem 4.1 implies

(5.55)$$ \begin{align} h_{\Phi_2}=h(\omega_{\Phi_2}) =h\left(\omega_{\Phi_1}\circ\tau_{1,0}^{\Psi}\right)=h(\omega_{\Phi_1}) =h_{\Phi_1}, \end{align} $$

proving the stability.

6 Automorphisms with factorised $d^{0}_{H_{U}}\alpha $

When $\alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$ has some good factorisation property with respect to the action of $\beta _{g}^{U}$, the index $h(\omega )$ can be calculated without going through GNS representations.

Definition 6.1. For $\alpha \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}\right )$, we set

(6.1)$$ \begin{align} \left ( d^{0}_{H_{U}}\alpha\right ) (g):=\alpha^{-1}\beta_{g}^{U}\circ \alpha\circ \left ( \beta_{g}^{U}\right )^{-1},\quad g\in G. \end{align} $$

We say that $d^{0}_{H_{U}}\alpha $ is factorised into left and right if there are automorphisms $\gamma _{g,\sigma }\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{H_{\sigma }}\right )$, $g\in G, \sigma =L,R$, such that

(6.2)$$ \begin{align} \left ( d^{0}_{H_{U}}\alpha\right ) (g)=({\textrm{{inner}}})\circ \left ( \gamma_{g,L}\otimes\gamma_{g,R}\right ),\quad g\in G. \end{align} $$

For known examples of $2$-dimensional SPT phases like [CGLW, MM, Y, DW] or injective projected entangled-pair states [MGSC], this property holds. Namely, with a bit of effort, states in these models can be written in the form $\omega _0\alpha $, where $\omega _0$ is a pure infinite tensor product state and $\alpha $ is an automorphism satisfying the property in Definition 6.1. From such an automorphism, we can derive an outer action of G.

Lemma 6.2. Let $\alpha \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}\right )$ be an automorphism. Suppose that $d^{0}_{H_{U}}\alpha $ is factorised into left and right – that is, there are automorphisms $\gamma _{g,\sigma }\in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}_{H_{\sigma }}\right )$, $g\in G, \sigma =L,R$, such that

(6.3)$$ \begin{align} \left ( d^{0}_{H_{U}}\alpha\right ) (g)=({\textrm{{inner}}})\circ \left ( \gamma_{g,L}\otimes\gamma_{g,R}\right ),\quad g\in G. \end{align} $$

Then there are unitaries $v_{\sigma }(g,h)\in {\mathcal U}\left ( {\mathcal A}_{H_{\sigma }}\right )$, $g,h\in G, \sigma =L,R$, such that

(6.4)$$ \begin{align} \gamma_{g,{\sigma}}\beta_{g}^{{\sigma}U}\gamma_{h,{\sigma}}\beta_{h}^{{\sigma}U} \left ( \gamma_{gh,{\sigma}}\beta_{gh}^{{\sigma}U}\right )^{-1} =\mathop{\mathrm{Ad}}\nolimits( v_{\sigma}(g,h)). \end{align} $$

Proof. Because $\beta _{g}^{U}$ is a group action, substituting equation (6.3) we get

(6.5)$$ \begin{align} \mathop{\mathrm{id}}\nolimits_{{\mathcal A}} &=\alpha^{-1}\beta_{g}^{U}\alpha\circ\alpha^{-1}\beta_{h}^{U}\alpha\circ \left ( \alpha^{-1}\beta_{gh}^{U}\alpha\right )^{-1}\notag\\ &=({\textrm{{inner}}})\circ \left ( \gamma_{g,L}\beta_{g}^{LU}\otimes\gamma_{g,R}\beta_{g}^{RU}\right ) \circ \left ( \gamma_{h,L}\beta_{h}^{LU}\otimes\gamma_{h,R}\beta_{h}^{RU}\right ) \circ \left ( \gamma_{gh,L}\beta_{gh}^{LU}\otimes\gamma_{gh,R}\beta_{gh}^{RU}\right )^{-1}\notag\\ &=({\textrm{{inner}}})\circ \left ( \gamma_{g,L}\beta_{g}^{LU}\gamma_{h,L}\beta_{h}^{LU} \left ( \gamma_{gh,L}\beta_{gh}^{LU}\right )^{-1} \otimes\gamma_{g,R}\beta_{g}^{RU} \gamma_{h,R}\beta_{h}^{RU}\left ( \gamma_{gh,R}\beta_{gh}^{RU}\right )^{-1}\right ). \end{align} $$

By Lemma B.1, we then see that there are unitaries $v_{\sigma }(g,h)\in \mathop {\mathrm {Aut}}\nolimits \left ( {\mathcal A}_{H_{\sigma }}\right )$, $g\in G, \sigma =L,R$, satisfying equation (6.4).

It is well known that a third cohomology class can be associated to cocycle actions [C, J].

Lemma 6.3. Let $\alpha \in \mathop {\mathrm {Aut}}\nolimits \left ({\mathcal A}\right )$ be an automorphism such that $d^{0}_{H_{U}}\alpha $ is factorised into left and right as in equation (6.3). Let $v_{\sigma }(g,h)\in {\mathcal U}\left ( {\mathcal A}_{H_{\sigma }}\right )$, $g,h\in G, \sigma =L,R$, be unitaries satisfying equation (6.4). Then there is some $c_{{\sigma }}\in C^{3}(G,{\mathbb T})$, $\sigma =L,R$, such that

(6.6)$$ \begin{align} v_{{\sigma}}(g,h) v_{{\sigma}}(gh,k) =c_{{\sigma}}(g,h,k) \left ( \gamma_{g,{\sigma}}\circ\beta_{g}^{{\sigma}U}\left ( v_{{\sigma}}(h,k)\right )\right ) v_{{\sigma}}\left ( g, hk\right ),\quad g,h,k\in G. \end{align} $$

Proof. By equation (6.4), we have

(6.7)$$ \begin{align}\hat\gamma_{g,\sigma}\hat\gamma_{h,\sigma} =\mathop{\mathrm{Ad}}\nolimits( v_{\sigma}(g,h))\circ \hat\gamma_{gh,\sigma} \end{align} $$

for $ \hat \gamma _{g,\sigma }:=\gamma _{g,{\sigma }}\beta _{g}^{{\sigma }U} $. Using this, we have

(6.8)$$ \begin{align} &\mathop{\mathrm{Ad}}\nolimits( v_{\sigma}(g,h))\circ\mathop{\mathrm{Ad}}\nolimits\left ( v_{\sigma}(gh,k)\right )\circ\hat\gamma_{ghk,\sigma} \notag \\ &\quad =\mathop{\mathrm{Ad}}\nolimits( v_{\sigma}(g,h))\circ \hat\gamma_{gh,\sigma}\circ\hat\gamma_{k,\sigma} =\hat\gamma_{g,\sigma}\hat\gamma_{h,\sigma}\hat\gamma_{k,\sigma} =\hat\gamma_{g,\sigma}\circ \mathop{\mathrm{Ad}}\nolimits\left ( v_{\sigma}(h,k)\right )\circ\hat\gamma_{hk,\sigma}\notag\\ &\quad =\mathop{\mathrm{Ad}}\nolimits\left ( \hat\gamma_{g,\sigma}\left ( v_{\sigma}(h,k)\right )\right ) \hat\gamma_{g,\sigma}\circ \hat\gamma_{hk,\sigma} =\mathop{\mathrm{Ad}}\nolimits\left ( \hat\gamma_{g,\sigma}\left ( v_{\sigma}(h,k)\right ) v_{\sigma}(g,hk) \right )\circ\hat\gamma_{ghk,\sigma}. \end{align} $$

Because ${\mathcal A}'\cap {\mathcal A}=\mathbb I_{{\mathcal A}}$, it must be the case that $\hat \gamma _{g,\sigma }\left ( v_{\sigma }(h,k)\right ) v_{\sigma }(g,hk)$ and $v_{\sigma }(g,h)v_{\sigma }(gh,k)$ are proportional to each other, proving the lemma.

By the same argument as in Lemma 2.4, we can show that this $c_{R}$ is actually a $3$-cocycle. If $\omega \in \mathop {\mathcal {SL}}\nolimits $ is given by an automorphism $\alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega )$ with factorised $d^{0}_{H_{U}}\alpha $, and if $\omega _{0}$ is invariant under $\beta _{g}^{U}$, then we have $h(\omega )=[c_{R}]_{H^{3}\left (G,{\mathbb T}\right )}$, for $c_R$ given in Lemma 6.3.

Theorem 6.4. Let $\omega _{0}$ be a reference state of the form in formula (2.18), and assume that $\omega _{0}\circ \beta _{g}^{U}=\omega _{0}$ for any $g\in G$. Let $\alpha \in \mathop {\mathrm {QAut}}\nolimits \left ( {\mathcal A}\right )$ be an automorphism. Suppose that $d^{0}_{H_{U}}\alpha $ is factorised into left and right as in equation (6.3) with some $\gamma _{g,\sigma }\in \mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{C_{\theta _{0}},\sigma } )$ and $0<\theta _{0}<\frac \pi 2$, for $\sigma =L,R$. Let $v_{\sigma }(g,h)\in {\mathcal U}\left ( {\mathcal A}_{H_{\sigma }}\right )$, $g,h\in G, \sigma =L,R$, be unitaries satisfying equation (6.4) and $c_{R}\in C^{3}(G,{\mathbb T})$ satisfying equation (6.6) for these $v_{R}(g,h)$ which are given in Lemma 6.2 and Lemma 6.3. Then we have $\omega _{0}\circ \alpha \in \mathop {\mathcal {SL}}\nolimits $ with $\mathop {\mathrm {IG}}\nolimits (\omega _{0}\circ \alpha )\neq \emptyset $, $c_{R}\in Z^{3}(G,{\mathbb T})$, and $h(\omega _{0}\circ \alpha )=[c_{R}]_{H^{3}\left (G,{\mathbb T}\right )}$.

Remark 6.5. The situation of this theorem is special. We do not expect that it always occurs.

Proof. That $\omega _{0}\circ \alpha \in \mathop {\mathcal {SL}}\nolimits $ is by definition. Because

(6.9)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( v_{\sigma}(g,h)) = \gamma_{g,{\sigma}}\beta_{g}^{{\sigma}U}\gamma_{h,{\sigma}}\beta_{h}^{{\sigma}U} \left ( \gamma_{gh,{\sigma}}\beta_{gh}^{{\sigma}U}\right )^{-1} \in \mathop{\mathrm{Aut}}\nolimits\left ({\mathcal A}_{C_{\theta_{0}},\sigma}\right ), \end{align} $$

our $ v_{\sigma }(g,h)$ belongs to ${\mathcal U}({\mathcal A}_{C_{\theta _{0}},\sigma } )$. Because

(6.10)$$ \begin{align} \omega_{0}\alpha\circ\alpha^{-1}\beta_{g}^{U}\alpha =\omega_{0}\beta_{g}^{U}\alpha =\omega_{0}\alpha \end{align} $$

and

(6.11)$$ \begin{align} \alpha^{-1}\beta_{g}^{U}\alpha =({\textrm{{inner}}})\circ \left ( \gamma_{g,L}\otimes\gamma_{g,R}\right )\circ\beta_{g}^{U}, \end{align} $$

with $\gamma _{g,\sigma }\in \mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{C_{\theta _{0},\sigma }} )$, we have $(\alpha ^{-1}\beta _{g}^{U}\alpha )\in \mathop {\mathrm {IG}}\nolimits (\omega _{0}\alpha , \theta _{0})$, and $ (\gamma _{g,\sigma } )\in {\mathcal T}(\theta _{0}, \alpha ^{-1}\beta _{g}^{U}\alpha )$. Clearly $\alpha \in \mathop {\mathrm {EAut}}\nolimits (\omega _{0}\circ \alpha )$, and there is $(\alpha _L,\alpha _R,\Theta )\in {\mathcal D}_{\alpha }^{\theta _{0}}$ because $\alpha \in \mathop {\mathrm {QAut}}\nolimits ({\mathcal A})$. Set $\gamma _{g}:=\gamma _{g,L}\otimes \gamma _{g,R}$. From Lemma 2.1, there is some $W_{g}\in {\mathcal U}({\mathcal H}_{0}) g\in G$ satisfying

(6.12)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g)\circ\pi_0 =\pi_0\circ\left ( \alpha_L\otimes\alpha_{R}\right ) \circ\Theta\circ\gamma_g\beta_g^U\circ\Theta^{-1}\circ \left ( \alpha_L\otimes\alpha_{R}\right )^{-1},\quad g\in G. \end{align} $$

In particular, because $ v_{R}(h,k)$ belongs to ${\mathcal U}({\mathcal A}_{( C_{\theta _{0}})_{R}} )$, $\Theta \in \mathop {\mathrm {Aut}}\nolimits ({\mathcal A}_{C_{\theta _{0}}^{c}})$, and $\gamma _g\beta _g^U$ preserves ${\mathcal A}_{\left ( C_{\theta _{0}}\right )_{R}}$, we have

(6.13)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits( W_g)&\circ \pi_0\circ\left ( \alpha_L\otimes\alpha_{R}\right ) \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{{L}}}\otimes \left ( v_{R}(h,k)\right )\right ) \notag \\ &=\pi_0\circ\left ( \alpha_L\otimes\alpha_{R}\right ) \circ\Theta\circ\gamma_g\beta_g^U\circ\Theta^{-1}\circ \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{{L}}}\otimes \left ( v_{R}(h,k)\right )\right )\notag\\ &=\pi_0\circ\left ( \alpha_L\otimes\alpha_{R}\right ) \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal A}_{{L}}}\otimes \gamma_{g,R}\beta_g^{RU}\left ( v_{R}(h,k)\right )\right ) =\mathbb I_{{\mathcal H}_{L}}\otimes \pi_R\circ \alpha_{R}\circ \gamma_{g,R}\beta_g^{RU}( v_{R}(h,k) ). \end{align} $$

On the other hand, equation (6.4) means

(6.14)$$ \begin{align} \mathop{\mathrm{Ad}}\nolimits\left ( \pi_{\sigma}\circ\alpha_{\sigma}( v_{\sigma}(g,h)) \right )\pi_{\sigma} =\pi_{\sigma}\circ\alpha_{\sigma}\circ \gamma_{g,{\sigma}}\beta_{g}^{{\sigma}U}\gamma_{h,{\sigma}}\left ( \beta_{g}^{{\sigma}U}\right )^{-1} ( \gamma_{gh,{\sigma}})^{-1}\circ\alpha_{\sigma}^{-1}. \end{align} $$

From equations (6.12) and (6.14), we have

(6.15)$$ \begin{align} \left ( (W_g), (\pi_{\sigma}\circ\alpha_{\sigma}( v_{\sigma}(g,h)))\right )\in \mathop{\mathrm{IP}}\nolimits\left ( \omega_{0}\circ\alpha, \alpha, \theta_{0}, \left(\alpha^{-1}\beta_{g}^{U}\alpha\right), \left(\gamma_{g,\sigma}\right), (\alpha_L,\alpha_R,\Theta) \right ). \end{align} $$

Now from equations (6.6) and then (6.13), we obtain

(6.16)$$ \begin{align} &\mathbb{I}_{{\mathcal H}_{L}}\otimes\pi_{R}\circ\alpha_{R}\left ( v_{R}(g,h)v_{R}(gh,k)\right ) \notag \\ &\quad =c_{{R}}(g,h,k) \mathbb I_{{\mathcal H}_{L}}\otimes\pi_{R}\circ\alpha_{R}\left ( \left ( \gamma_{g,{R}}\circ\beta_{g}^{{R}U}\left ( v_{{R}}(h,k)\right )\right ) v_{{R}}( g, hk ) \right)\notag\\ &\quad =c_{{R}}(g,h,k) \left ( \mathop{\mathrm{Ad}}\nolimits( W_g) \left ( \mathop{\mathrm{id}}\nolimits_{{\mathcal H}_{{L}}}\otimes \pi_{R}\alpha_{R}\left ( v_{R}(h,k)\right )\right )\right ) \cdot \left ( \mathbb I_{{\mathcal H}_{L}}\otimes\pi_{R}\circ\alpha_{R}\left ( v_{{R}}\left ( g, hk\right )\right )\right ). \end{align} $$

This means

(6.17)$$ \begin{align} c_{R}=c_R\left ( \omega_{0}\circ\alpha, \alpha, \theta_{0}, \left(\alpha^{-1}\beta_g^{U}\alpha\right), \left(\gamma_{g,\sigma}\right), (\alpha_L,\alpha_R,\Theta),\left ( (W_g), ((\pi_{\sigma}\circ\alpha_{\sigma}( v_{\sigma}(g,h))))\right ) \right ) \end{align} $$

in Definition 2.5. Hence we get $c_{R}\in Z^{3}(G,{\mathbb T})$, and $h(\omega _{0}\circ \alpha )=[c_{R}]_{H^{3}\left (G,{\mathbb T}\right )}$.