$C^*$-ALGEBRAS OF SELF-SIMILAR ACTION OF GROUPOIDS ON ROW-FINITE DIRECTED GRAPHS

Abstract

For amenable discrete groupoids $\mathcal {G}$ and row-finite directed graphs E, let $(\mathcal {G},E)$ be a self-similar groupoid and let $C^*(\mathcal {G}, E)$ be the associated $C^*$-algebra. We introduce a weaker faithfulness condition than those in the existing literature that still guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$. Under this faithfulness condition, we prove a gauge-invariant uniqueness theorem.

1 Introduction

Roughly speaking, if parts of an object are similar to the whole, repeating the structure of the object at all scales, then we call the object self-similar. If a group or a groupoid acts self-similarly on a space, then we simply call it a self-similar group or a self-similar groupoid. Self-similar groups were introduced by Grigorchuk in [2] and Gupta and Sidki in [3] to answer the question of existence of groups with intermediate growth. Recently, operator algebraists have made use of self-similar groups to study $C^*$ -algebras (for example, [1, 4]). Since a groupoid is a generalisation of a group, it is then natural to think of this notion of self-similarity on a groupoid, as introduced in [5]. Self-similar groups act on the path-spaces of graphs with a single vertex. To study self-similar actions on more general directed graphs and the associated Cuntz–Krieger algebras, Laca et al. in [5] introduced the notion of a self-similar groupoid. In [5], the authors are primarily interested in computing KMS states, so, informed by results about graph $C^*$ -algebras, they restricted their attention to finite graphs. They also built their self-similar groupoids by generalising the process whereby automata are used to build self-similar groups, so by definition their self-similar actions satisfy a faithfulness condition that simplifies their analysis and, in particular, guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$ .

Another approach to self-similar actions on graphs with multiple vertices was developed by Exel and Pardo [1] and does not require a faithfulness condition. We combine and generalise the constructions in [1, 5]. We consider self-similar actions of groupoids $\mathcal {G}$ on the path spaces $E^*$ of row-finite directed graphs E that are not necessarily faithful in the sense of [5]. We develop a new faithfulness condition that is weaker than both faithfulness as in [5] and pseudo-faithfulness as in [1], but still guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$ , and we prove a gauge-invariant uniqueness theorem. In particular, our theorems apply to conventional actions of groups on graphs (see Example 3.7). We also depart from [5] in that we work solely with generators and relations, without employing the machinery of Hilbert modules and Cuntz–Pimsner algebras.

The paper is organised as follows. We define our notion of a self-similar groupoid $(\mathcal {G}, E)$ in Definition 2.1 and construct the associated $C^*$ -algebras $C^*(\mathcal {G}, E)$ in Section 3 following the approach of [6]. We introduce our injectivity condition in our key technical result Proposition 3.6. We analyse the fixed-point algebra $C^*(\mathcal {G},E)^{\gamma }$ for the gauge action $\gamma $ in Section 4. By applying all the results in the previous sections, we prove the gauge-invariant uniqueness theorem in Theorem 5.1.

2 Self-similar groupoids

Recall that a groupoid $\mathcal {G}$ is a small category with inverses. We write $\mathcal {G}^{(0)}$ for the set of identity morphisms and $r, s : \mathcal {G} \rightarrow \mathcal {G}^{(0)}$ for the maps induced by the codomain and domain range maps. Throughout this paper, $\mathcal {G}$ will denote a countable discrete groupoid. We will assume that $\mathcal {G}$ is amenable in the sense of [7]. Since $\mathcal {G}$ is discrete, this is equivalent to requiring that its full and reduced $C^*$ -algebras coincide, and is also equivalent to requiring that each of its isotropy groups is amenable.

As in [6], a (directed) graph is a quadruple $E=(E^0, E^1, r, s)$ consisting of countable sets $E^0$ , $E^1$ and maps $r, s: E^1 \rightarrow E^0$ . Elements of $E^1$ are called edges and elements of $E^0$ are called vertices. We will assume that all our graphs are row-finite and have no sources in the sense that $0 < \lvert r^{-1}(v) \rvert < \infty $ for all $v \in E^0$ . Let $e, f \in E^1$ with $s(e)=r(f)$ . Then, $ef$ is a path of length 2 and we write $\lvert ef \rvert =2$ . In general, a path $\mu $ of length n in E is a sequence $\mu _1\mu _2 \cdots \mu _n$ such that $s(\,\mu _i)=r(\,\mu _{i+1})$ for $1\leq i \leq n-1$ . The vertices are viewed as paths of length 0. The paths of length n are collected in a set denoted by $E^n$ . We let $E^*:=\bigcup _{k\geq 0} E^k$ . It is natural to extend the maps $r,s$ to $E^*$ by putting $r(\,\mu )=r(\,\mu _1)$ and $s(\,\mu )=s(\,\mu _{\lvert \,\mu \rvert })$ where $\lvert \,\mu \rvert>1$ , and $r(v)=v=s(v)$ for $v \in E^0$ .

Definition 2.1. Let E be a row-finite graph with no sources and let $\mathcal {G}$ be a groupoid with $\mathcal {G}^{(0)}=E^0$ . Write

$$ \begin{align*} \mathcal{G}\ast E^*:= \{(g, \mu) \in \mathcal{G}\times E^* \mid s(g)=r(\,\mu)\} \end{align*} $$

and

$$ \begin{align*} E^* \ast \mathcal{G} := \{(\,\mu, g) \mid s(\,\mu)=r(g)\}. \end{align*} $$

We will often denote the element $(\,\mu , g)\in E^* \ast \mathcal {G}$ by the shorthand $\mu g$ . A self-similar action of $\mathcal {G}$ on $E^*$ consists of two maps: (1) an action $(g,\mu )\mapsto g\cdot \mu $ of $\mathcal {G}$ on the set $E^*$ and (2) a map $\varphi : \mathcal {G}\ast E^* \rightarrow \mathcal {G}$ such that:

1. (i) $g\cdot (\kern1pt\mu \beta )=(g\cdot \mu )(\varphi (g, \mu )\cdot \beta )$ ;

2. (ii) $r(g\cdot \mu )=g\cdot r(\,\mu )$ and $s(g\cdot \mu )=\varphi (g, \mu ) \cdot s(\,\mu )$ ;

3. (iii) $\lvert g\cdot \mu \rvert =\lvert \,\mu \rvert $ ;

4. (iv) $\varphi (g, v)=g$ ;

5. (v) $\varphi (gh, \mu )=\varphi (g, h\cdot \mu )\varphi (h, \mu )$ ;

6. (vi) $\varphi (g, \mu \beta )=\varphi (\varphi (g,\mu ),\beta )$ ; and

7. (vii) $\varphi (g^{-1}, \mu )=(\varphi (g, g^{-1}\cdot \mu ))^{-1}$ .

We write this self-similar action of the groupoid $\mathcal {G}$ on $E^*$ as a pair $(\mathcal {G},E)$ and call it a self-similar groupoid $(\mathcal {G},E)$ .

3 The universal $C^*$ -algebra $C^*(\mathcal {G}, E)$

Recall that a Toeplitz–Cuntz–Krieger family for a row-finite directed graph E with no sources consists of partial isometries $\{T_e \mid e \in E^1\}$ and mutually orthogonal projections $\{W_v \mid v \in E^0\}$ satisfying $T_e^*T_e=W_{s(e)}$ and $W_v \geq \sum _{e \in vE^1}T_eT^*_e$ for all $v \in E^0$ . It is a Cuntz–Krieger E-family if $W_v=\sum _{e\in vE^1}T_eT^*_e$ for all $v \in E^0$ . A unitary representation in a unital $C^*$ -algebra A of a discrete groupoid $\mathcal {G}$ is a family $\{U_g \mid g \in \mathcal {G}\}$ of partial isometries such that $U_gU_h=\delta _{s(g),r(h)}U_{gh}$ and $U_{g^{-1}}=U_g^*$ for all $g, h \in \mathcal {G}$ , and such that $\sum _{v\in \mathcal {G}^{(0)}}U_v=1_A$ .

Definition 3.1. Let $(\mathcal {G},E)$ be a self-similar groupoid. A Toeplitz $(\mathcal {G}, E)$ -family consists of partial isometries $\{T_e \mid e \in E^1\}$ and a unitary representation $\{W_g \mid g \in \mathcal {G}\}$ of $\mathcal {G}$ such that $\{T_e \mid e \in E^1\} \cup \{W_v \mid v \in E^0\} $ is a Toeplitz–Cuntz–Krieger E-family. It is a Cuntz–Krieger $(\mathcal {G}, E)$ -family if $\{T_e, W_v\}$ is a Cuntz–Krieger E-family.

Example 3.2. Suppose that $\mathcal {G}$ acts self-similarly on E. Let $\mathcal {H}:=l^2(E^* \ast \mathcal {G})$ with orthonormal basis $\{e_{\mu g} \mid \mu \in E^*, g \in \mathcal {G} \}$ . For $e \in E^1$ and $h \in \mathcal {G}$ , let $T_e, W_h \in B(\mathcal {H})$ be the operators such that

$$ \begin{align*} T_e e_{\mu g} &= \begin{cases} e_{e \mu g} & \mathrm{if} \ s(e)=r(\,\mu), \\ 0 & \mathrm{otherwise}, \end{cases} \\[6pt] W_h e_{\mu g} &= \begin{cases} e_{(h \cdot \mu) (\varphi (h, \mu)\cdot g)} & \mathrm{if} \ s(h)= r(\,\mu), \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$

For $v \in E^0$ , $W_v$ is the projection onto $l^2(\{\,\mu g\mid r(\,\mu )=v\}) \subset \mathcal {H}$ , and a routine calculation shows that for $e \in E^1$ ,

$$ \begin{align*} T_e^* e_{\mu g} = \begin{cases} e_{\mu' g} & \mathrm{if} \ \mu=e\mu', \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$

It is routine to check that the family $\{T_e \mid e \in E^1\} \cup \{ W_h \mid h \in \mathcal {G}\}$ is a Toeplitz $(\mathcal {G}, E)$ -family in $B(\mathcal {H})$ .

The proofs of the following two lemmas are more or less identical to those of the cited results in [4–6].

Lemma 3.3 (See [4, Lemma 3.4] and [5, Lemma 4.6])

Let $(\mathcal {G},E)$ be a self-similar groupoid. Suppose that $\{T_e,W_g\}$ is a Toeplitz $(\mathcal {G}, E)$ -family in a $C^*$ -algebra B. Then for all $\mu , \beta , \alpha , \rho \in E^*$ , $g, h \in \mathcal {G}$ ,

$$ \begin{align*}(T_{\mu} W_g T^*_{\beta})(T_{\alpha} W_h T^*_{\rho})=\begin{cases} T_{\mu(g \cdot \alpha')} W_{\varphi(g,\alpha')h} T^{*}_{\rho} & \mathrm{if} \ \alpha=\beta \alpha', \\ T_{\mu} W_{g\varphi(h, h^{-1}\cdot \beta')} T^{*}_{\rho(h^{-1}\cdot \beta')} & \mathrm{if} \ \beta=\alpha\beta', \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$

Lemma 3.4 (See [6, Corollary 1.16])

Let $(\mathcal {G},E)$ be a self-similar groupoid. Suppose that $\{T_e,W_g\}$ is a Toeplitz $(\mathcal {G}, E)$ -family. Then

$$ \begin{align*}C^*(T,W)=\overline{\mathrm{span}}\{T_{\mu} W_g T^*_{\beta} \mid \mu, \beta \in E^*, g \in \mathcal{G}^{s(\,\mu)}_{s(\,\beta)}, s(\,\mu)=g\cdot s(\,\beta)\}.\end{align*} $$

A standard argument along the lines of Propositions 1.20 and 1.21 of [6] shows that there exists a $C^*$ -algebra $\mathcal {T}C^*(\mathcal {G}, E)$ generated by a Toeplitz $(\mathcal {G},E)$ -family $\{t_e, w_g\}$ that is universal in the sense that for any Toeplitz $(\mathcal {G},E)$ -family $\{T_e, W_g\}$ , there is a homomorphism $\pi _{T,W}: \mathcal {T}C^*(\mathcal {G}, E) \rightarrow C^*(T,W)$ such that $\pi _{T,W}(t_e)=T_e$ for all $e \in E^1$ and $\pi _{T,W}(w_g)=W_g$ for all $g \in \mathcal {G}$ .

Let I be the ideal of $\mathcal {T}C^*(\mathcal {G}, E)$ generated by $\{w_v-\sum _{r(e)=v}t_et^*_e \mid v \in E^0\}$ . Then $s_e:=t_e+I$ for all $e \in E^1$ and $u_g:=w_g+I$ for all $g \in \mathcal {G}$ defines a Cuntz–Krieger $(\mathcal {G}, E)$ -family and $C^*(\mathcal {G}, E):=\mathcal {T}C^*(\mathcal {G}, E)/I$ is universal for Cuntz–Krieger $(\mathcal {G}, E)$ - families. We will need to know that the generators of $C^*(\mathcal {G}, E)$ are nonzero. For this, we construct a concrete Cuntz–Krieger $(\mathcal {G}, E)$ -family (see Proposition 3.6).

Lemma 3.5. Let $(\mathcal {G}, E)$ be a self-similar groupoid. Let $\pi : C^*(T,W) \rightarrow B(l^2(E^* \ast \mathcal {G}))$ be the representation induced by the Toeplitz $(\mathcal {G}, E)$ -family $\{T_e, W_g\}$ of Example 3.2. For every $a \in I$ and every $\varepsilon>0$ , there exists $N \in \mathbb {N}$ such that for all $n \geq N$ ,

$$ \begin{align*} \lVert\pi(a)|_{\overline{\mathrm{span}}\{e_{\lambda g} \mid \lambda \in E^n, g \in \mathcal{G}^{s(\lambda)}\}}\rVert <\varepsilon. \end{align*} $$

Proof. First, note that for $v \in E^0, \lambda \in E^*$ and $k \in \mathcal {G}^{s(\lambda )}$ ,

(3.1) $$ \begin{align} \bigg(W_v-\sum_{e\in vE^1}T_eT_e^*\bigg ) e_{\lambda k} & = \begin{cases} 0 & \mathrm{if} \ \lambda \neq v, \\ e_{\lambda k} & \mathrm{otherwise}. \end{cases} \end{align} $$

Now fix $v \in E^0, \mu , \beta \in E^*$ and $g \in \mathcal {G}^{s(\,\mu )}_v, h \in \mathcal {G}^{s(\,\beta )}_v$ . Then,

$$ \begin{align*} T_{\mu} W_g & \bigg(W_v-\sum_{e\in vE^1} T_eT^*_e \bigg)W^*_h T^*_{\beta} e_{\lambda k} \\ &= \begin{cases} T_{\mu} W_g (W_v-\sum_{e\in vE^1} T_eT^*_e) e_{(h^{-1} \cdot \lambda^{'}) \varphi(h^{-1}, \lambda^{'})k} & \mathrm{if} \ \lambda=\beta \lambda^{'}, \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$

By (3.1), this equals $0$ if $\lvert \lambda ^{'}\rvert>0$ . Hence,

(3.2) $$ \begin{align} \bigg\lVert T_{\mu} W_g \bigg (W_v-\sum_{e\in vE^1} T_eT^*_e\bigg )W^*_h T^*_{\beta} e_{\lambda k}\bigg \rVert = 0 \quad \textrm{whenever} \ \lvert \lambda\rvert> \lvert \,\beta\rvert. \end{align} $$

Fix a finite linear combination $a_0=\sum _{a_{\mu , g, h, \beta }}T_{\mu } W_g(W_v-\sum _{e\in vE^1}T_eT^*_e)W_h^*T_{\beta }^*$ . Let $N=\mathrm { \max }\{\lvert \,\beta \rvert \mid a_{\mu ,g,h,\beta }\neq 0\}$ . Then (3.2) implies that $\lVert a_0 e_{\lambda k}\rVert = 0$ whenever $\lvert \lambda \rvert> N$ .

Finally, fix $a \in I$ , and $\varepsilon>0$ . A routine argument gives

$$ \begin{align*} I=\overline{\mathrm{span}}\bigg \{t_{\mu} w_g \bigg (w_v-\sum_{e\in vE^1}t_et_e^*\bigg )w_h^*t^*_{\beta} \mid \mu, \beta \in E^*, g,h \in \mathcal{G}, v \in E^0\bigg \}. \end{align*} $$

So there exists

$$ \begin{align*} a_0\in \mathrm{span}\bigg \{T_{\mu} W_g \bigg (W_v-\sum_{e\in vE^1}T_eT^*_e\bigg ) W^*_hT^*_{\beta} \mid \mu, \beta \in E^*, g, h \in \mathcal{G}, v \in E^0 \bigg \}, \end{align*} $$

such that $\lVert \pi (a)-a_0\rVert <\varepsilon .$

Take N as above and fix $n \geq N$ . Then,

$$ \begin{align*} \lVert \pi(a)|_{\overline{\mathrm{span}}\{e_{\lambda k} \mid \lambda\in E^n, k\in \mathcal{G}^{s(\lambda)}\}}\lVert \leq \rVert \pi(a)-a_0\lVert + \rVert a_0|_{\overline{\mathrm{span}}\{e_{\lambda k} \mid \lambda\in E^n, k\in \mathcal{G}^{s(\lambda)}\}}\rVert <\varepsilon.\\[-34pt] \end{align*} $$

The following proposition will be used in describing our fixed-point algebra in the next section. Let G be a discrete group and let $\mathcal {H}=l^2(G)=\overline {\mathrm {span}}\{\delta _g \mid g \in G\}$ . For $g \in G$ , define $\lambda _g \in \mathcal {U}(l^2(G))$ by $\lambda _g(\delta _h)=\delta _{gh}$ for all $h \in G$ . We get a representation $\lambda : C^*(G)\rightarrow B(\mathcal {H})$ such that $\lambda (u_g)=\lambda _g$ for all $g\in G$ ; we call this the regular representation. If G is amenable, then the representation $\lambda $ is faithful. Since our groupoid is an amenable (discrete) groupoid, its (discrete) isotropy groups are also amenable.

Proposition 3.6. Let $(\mathcal {G},E)$ be a self-similar groupoid. Let $\{s_e, u_g\}$ be the universal Cuntz–Krieger $(\mathcal {G},E)$ -family in $C^*(\mathcal {G},E)$ . Then each $s_e$ and each $u_g$ is nonzero. Fix $v \in E^0$ . The universal property of $C^*(\mathcal {G}^v_v)$ gives a homomorphism $\pi _u: C^*(\mathcal {G}^v_v) \rightarrow C^*(\mathcal {G},E)$ such that $\pi _u(\delta _h)=u_h$ for all $h \in \mathcal {G}^v_v$ . Suppose that for each $k \in \mathbb {N}$ , there exists $\lambda \in vE^k$ such that the map $g\mapsto (g\cdot \lambda )\varphi (g,\lambda )$ is injective. Then $\pi _u$ is injective.

Proof. By the universal property, it suffices to construct a Cuntz–Krieger $(\mathcal {G},E)$ -family $\{S_e, U_g\}$ consisting of nonzero partial isometries. If $\{S_e, U_g\}$ is a Cuntz–Krieger $(\mathcal {G}, E)$ -family and each $U_v\neq 0$ , then $S_e \neq 0$ for all $ e\in E^1$ and $U_g \neq 0$ for all $g \in \mathcal {G}$ , because $U_{s(e)}=S_e^*S_e$ and $U_{s(g)}=U_g^*U_g$ . So, it suffices to construct a $(\mathcal {G},E)$ -family with $U_v \neq 0$ for all $v \in E^0$ .

Let $\{T_e, W_g\}$ be the Toeplitz $(\mathcal {G},E)$ -family of Example 3.2. For $v \in E^0$ , we have $W_v \cdot e_{\mu s(\,\mu )}=e_{\mu s(\,\mu )}$ for all $\mu \in vE^*$ . So,

$$ \begin{align*} \lVert W_v|_{\overline{\mathrm{span}}\{e_{\lambda g} \mid \lambda\in vE^n, g\in \mathcal{G}^{s(\lambda)}\}}\rVert = 1. \end{align*} $$

Thus, Lemma 3.5 gives $W_v\not \in I$ . Therefore, $S_e:=T_e + I \ \mathrm {and} \ U_g:=W_g + I$ is a $(\mathcal {G}, E)$ -family with each $U_v \neq 0$ .

Now fix $v \in E^0$ . Let $\pi _W: C^*(\mathcal {G}^v_v) \rightarrow B(l^2(E^* \ast \mathcal {G}))$ be the homomorphism such that $\pi _W(\delta _h)=W_h$ . Fix $k\in \mathbb {N}$ . Choose $\lambda \in vE^*$ such that the map $h\mapsto (h\cdot \lambda )\varphi (h,\lambda )$ is injective. Let $\mathcal {H}_{\lambda }:=\overline {\mathrm {span}} \{e_{(h\cdot \lambda ) \varphi (h,\lambda )} \mid h \in \mathcal {G}^v_v\} \subseteq l^2(E^*\ast \mathcal {G})$ . By construction, $\mathcal {H}_{\lambda }$ is invariant for $\pi _W$ .

Since the map $g\mapsto (g\cdot \lambda )\varphi (g,\lambda )$ is injective, there is an inner-product preserving map $\phi _{\lambda } : l^2(\mathcal {G}^v_v) \rightarrow \mathcal {H}_{\lambda }$ that maps the element $e_g$ of the orthonormal basis of $l^2(\mathcal {G}^v_v)$ to the element $e_{(g \cdot \lambda ) \varphi (g, \lambda )}$ of the orthonormal basis of $\mathcal {H}_{\lambda }$ . For $h \in \mathcal {G}^v_v$ , define $V^{\lambda }_h \in \mathcal {U}(l^2(\mathcal {G}^v_v))$ by $V^{\lambda }_h =\phi ^*_{\lambda } W_h \phi _{\lambda } $ . We get

$$ \begin{align*} V^{\lambda}_h e_g = \phi^*_{\lambda} W_h \phi_{\lambda} e_g = \phi^*_{\lambda} W_h e_{(g \cdot \lambda)(\varphi(g, \lambda))} &=\phi^*_{\lambda} e_{(h \cdot (g \cdot \lambda))(\varphi(h, g \cdot \lambda)\varphi(g, \lambda))} \nonumber \\ &=\phi^*_{\lambda} e_{((hg) \cdot \lambda)(\varphi(hg, \lambda))}= e_{hg}. \nonumber \end{align*} $$

Hence, $\{V^{\lambda }_h \mid h \in \mathcal {G}^v_v\}\subseteq B(l^2(\mathcal {G}^v_v))$ is the regular representation of $\mathcal {G}^v_v$ and induces a faithful representation of $C^*(\mathcal {G}^v_v)$ . Hence, the reduction of $\pi _W$ to $\mathcal {H}_{\lambda }$ is injective, so its reduction to $l^2(E^k \ast \mathcal {G})$ is injective. Since k was arbitrary, the reduction of $\pi _W$ to $l^2(E^k \ast \mathcal {G})$ is injective, and hence isometric for all k.

Now, fix $a \in C^*(\mathcal {G}^v_v)\setminus \{0\}$ . Then for all k,

$$ \begin{align*} \lVert \pi_W(a)|_{\overline{\mathrm{span}}\{e_{\mu g} \mid \mu \in E^k, g \in \mathcal{G}^{s(\,\mu)}\}}\rVert = \lVert a\rVert \neq 0. \end{align*} $$

Thus, Lemma 3.5 implies $a \not \in I$ . We have $\pi _u(a)=a+I\neq 0$ . Therefore, the homomorphism $\pi _u$ is injective.

To see that our faithfulness condition is strictly weaker than that of [5], we provide the following example.

Example 3.7. Let E be the graph with one vertex and n edges $e_0, \ldots , e_{n-1}$ and let $\mathcal {G}=\mathbb {Z}$ . Define an action of $\mathcal {G}$ on E by $m \cdot e_i=e_{i+m}$ where addition is mod n, and define $\varphi (m, e_i)=m$ for all m. Then $\mathcal {G}$ does not act faithfully in the sense of [5], because $n \cdot e_i=e_i$ for all i. However, the map $(m, \lambda ) \mapsto (m \cdot \lambda , \varphi (m, \lambda ))$ is injective for each $\lambda $ because $\varphi (m, \lambda )=m$ and then $\lambda =\varphi (m,\lambda )^{-1} \cdot (m \cdot \lambda )$ . It is routine to see using universal properties that $C^*(\mathcal {G}, E)= \mathcal {O}_n \rtimes \mathbb {Z}$ .

4 The gauge action and the core

Let $\{s_e, u_g\}$ be the universal Cuntz–Krieger $(\mathcal {G},E)$ -family in $C^*(\mathcal {G}, E)$ . Then for $z \in \mathbb {T}$ , the family $\{zs_e, u_g\}$ is also a Cuntz–Krieger $(\mathcal {G},E)$ -family. So, the universal property gives a homomorphism $\gamma _z:C^*(\mathcal {G}, E) \rightarrow C^*(\mathcal {G}, E)$ such that $\gamma _z(s_e)=zs_e$ and $\gamma _z(u_g)=u_g$ for all $e,g$ . Since $\gamma _1$ agrees with the identity and $\gamma _z \circ \gamma _w$ agrees with $\gamma _{zw}$ on generators, $z \mapsto \gamma _z$ is an action. A standard $\varepsilon /3$ argument shows that it is a strongly continuous action, which we call the gauge action on $C^*(\mathcal {G}, E)$ . The fixed-point algebra of $\gamma $ is the $*$ -subalgebra

$$ \begin{align*}C^*(\mathcal{G}, E)^{\gamma}:=\{a\in C^*(\mathcal{G},E) \mid \gamma_z(a)=a \ \mathrm{for \ all} \ z\in \mathbb{T}\}\end{align*} $$

of $C^*(\mathcal {G}, E)$ . The following corollary describes $C^*(\mathcal {G}, E)^{\gamma }$ concretely.

Corollary 4.1. Let $(\mathcal {G},E)$ be a self-similar groupoid and let $\Phi : C^*(\mathcal {G}, E) \rightarrow C^*(\mathcal {G},E)^{\gamma }$ be the conditional expectation, $\Phi (a)=\int _{\mathbb {T}} \gamma _z(a)\,dz$ . Then,

$$ \begin{align*} \Phi (s_{\mu} u_g s^*_{\beta})=\delta_{\lvert \,\mu \rvert, \lvert \,\beta \rvert}s_{\mu} u_g s_{\beta}^* \quad\mbox{for } \mu, \beta \in E^* \mbox{ and } g\in \mathcal{G}_{s(\,\beta)}^{s(\,\mu)}. \end{align*} $$

Further, $C^*(\mathcal {G}, E)^{\gamma }=\overline {\mathrm {span}}\{s_{\mu } u_g s^*_{\beta } \mid s(\,\mu )=g\cdot s(\,\beta ) \ \mathrm {and} \ \lvert \,\mu \rvert =\lvert \,\beta \rvert \}. $

Proof. We have $\gamma _z(s_{\mu } u_g s^*_{\beta })=z^{ \lvert \,\mu \rvert -\lvert \,\beta \rvert }s_{\mu } u_g s^*_{\beta }$ , so $\Phi (s_{\mu } u_g s^*_{\beta }) = \delta _{\lvert \,\mu \rvert , \lvert \,\beta \rvert } s_{\mu } u_g s^*_{\beta }$ . Moreover, $\Phi (C^*(\mathcal {G},E))=\overline {\mathrm {span}}\{s_{\mu } u_g s^*_{\beta } \mid s(\,\mu )=g\cdot s(\,\beta ) \ \mathrm {and} \ \lvert \,\mu \rvert =\lvert \,\beta \rvert \}.$ Proposition 3.2 of [6] shows that $\Phi (C^*(\mathcal {G},E))=C^*(\mathcal {G},E)^{\gamma }$ .

Let $(\mathcal {G},E)$ be a self-similar groupoid and let $\{S_e, U_g\}$ be a Cuntz–Krieger $(\mathcal {G}, E)$ -family. For $k\in \mathbb {N}$ , we define

$$ \begin{align*} \mathcal{F}_k(S, U):=\overline{\mathrm{span}}\{S_{\mu} U_g S^*_{\beta} \mid \mu, \beta \in E^k, g \in \mathcal{G}^{s(\,\mu)}_{s(\,\beta)}, s(\,\mu)=g\cdot s(\,\beta) \}. \end{align*} $$

We define a relation $\sim $ on $E^0$ by $v\sim w$ if and only if $\mathcal {G}^v_w\neq \emptyset $ . Then $\sim $ is an equivalence relation. For $\xi \in E^0/{\sim }$ , define

$$ \begin{align*} \mathcal{F}_k (S, U, \xi):=\overline{\mathrm{span}}\{S_{\mu} U_g S^*_{\beta} \mid \mu, \beta \in E^k,g \in \mathcal{G}^{s(\,\mu)}_{s(\,\beta)}, s(\,\mu)=g\cdot s(\,\beta)\in \xi\}. \end{align*} $$

When $\{S_e, U_g\}$ is the universal family $\{s_e, u_g\}$ in $C^*(\mathcal {G}, E)$ , we write $\mathcal {F}_k:=\mathcal {F}_k(s,u)$ and $\mathcal {F}_k(\xi ):=\mathcal {F}_k(s, u, \xi )$ .

Notation 4.2. For the next few results, fix a self-similar groupoid $(\mathcal {G}, E)$ , an element $\xi \in E^0/\sim $ , a vertex $v \in \xi $ and for each $u \in \xi $ , an element $g_u \in \mathcal {G}^u_v$ (take $g_v =v$ ). We call $\{g_u\mid u\in \xi \}$ a spanning tree for $\mathcal {G}|_{\xi }$ . We denote $E^k \xi :=\{\,\mu \in E^k \mid \ s(\,\mu ) \in \xi \}$ .

Proposition 4.3. With Notation 4.2, let $\{S_e, U_g\}$ be a Cuntz–Krieger $(\mathcal {G}, E)$ -family. For $h \in \mathcal {G}^v_v$ and $\mu \in E^*$ , define $V_{h,\mu }:=S_{\mu } U_{g_{s(\,\mu )}}U_h(S_{\mu } U_{g_{s(\,\mu )}})^*$ . For each $k \in \mathbb {N}$ , the series $\sum _{\mu \in E^k \xi } V_{h,\mu }$ converges strictly to a partial unitary $\overline {V}_h$ in $\mathcal {M}C^*(S, U)$ and $\overline {V}_h \mathcal {F}_k(S, U, \xi )\subseteq \mathcal {F}_k(S, U, \xi )$ .

Proof. Fix $h \in \mathcal {G}^v_v$ . For $\mu \in E^k\xi $ ,

(4.1) $$ \begin{align} V_{h,\mu}V_{h,\mu}^* = S_{\mu} S^*_{\mu}= V_{h,\mu}^*V_{h,\mu}. \end{align} $$

For $\mu \neq \beta \in E^k\xi $ , $S_{\mu }S^*_{\mu }S_{\beta }S^*_{\beta }=0$ . So, $S^*_{\mu }S_{\beta }=0$ . Therefore, for $F\subseteq E^k\xi $ finite,

$$ \begin{align*} \bigg(\sum_{\mu \in F} V_{h,\mu}\bigg)\bigg(\sum_{\beta \in F} V_{h,\beta}\bigg)^*=\sum_{\mu \in F} V_{h,\mu} V^*_{h,\mu} =\sum_{\mu \in F}S_{\mu}S^*_{\mu}. \end{align*} $$

Now, fix $a\in C^*(S, U)$ . Then $a= \lim _K \sum _{v\in K}P_v a$ , where K ranges over all finite subsets of $E^0$ . Let $P_K =\sum _{v \in K}P_v$ . Fix $\varepsilon>0$ . There exists a finite set $K' \subseteq E^0$ such that $\lVert P_Ka-a \rVert <\varepsilon /2$ for all finite $K \supseteq K'$ .

Let $F \subseteq E^k \xi $ be the finite set $F=KE^k\xi $ . For $F', F" \supseteq F$ ,

$$ \begin{align*} \bigg \lVert \sum_{\mu \in F'}S_{\mu} S^{*}_{\mu} a- \sum_{\beta \in F"}S_{\beta} S^{*}_{\beta} a\bigg \rVert & \leq \bigg\lVert \sum_{\mu \in F' \setminus F"}S_{\mu} S^*_{\mu} a \bigg\rVert + \bigg\lVert\sum_{\beta \in F"\setminus F'}S_{\beta} S^*_{\beta} a \bigg\rVert \\ \nonumber & \leq \lVert (1-P_K)a\rVert + \lVert(1-P_K)a\rVert < \varepsilon. \nonumber \end{align*} $$

So, $(\sum _{\mu \in F}S_{\mu } S^*_{\mu } a )_{F \subseteq E^k \xi }$ is Cauchy and hence converges. Thus, $\sum _{\mu \in E^k \xi } S_{\mu } S^*_{\mu }$ converges strictly to a projection $P_{\xi } \in \mathcal {M}C^*(S, U)$ . Equation (4.1) shows that $\sum _{\mu \in F} V^*_{h,\mu }V_{h,\mu }$ also converges strictly to $P_{\xi }$ . Therefore, $\sum _{\mu \in E^k\xi } V_{h,\mu }$ converges strictly to a unitary $\overline {V}_h \in P_{\xi }\mathcal {M}C^*(S, U)P_{\xi }$ .

Now fix a spanning element $S_{\alpha } U_l S^*_{\beta }$ of $\mathcal {F}_k(S, U, \xi )$ . For each $\mu \in E^k\xi $ , we obtain

$$ \begin{align*} V_{h, \mu}S_{\alpha} U_l S^*_{\beta} &=\delta_{\mu, \alpha}S_{\mu}U_{g'l} S^*_{\beta} \quad\text{for some} \ g'\in\mathcal{G}^{s(\,\mu)}_{s(\,\mu)}, \end{align*} $$

which implies that

$$ \begin{align*} \overline{V}_h S_{\alpha} U_l S^*_{\beta} =\sum_{\mu \in E^k \xi} \delta_{\mu, \alpha} S_{\mu} U_{g'l} S^*_{\beta}& = S_{\alpha} U_{g'l} S^*_{\beta} \quad \text{for some} \ g'\in\mathcal{G}^{s(\alpha)}_{s(\alpha)},\\ & \in \mathcal{F}_k(S, U, \xi). \end{align*} $$

Hence, $\overline {V}_h \mathcal {F}_k(S, U, \xi )\subseteq \mathcal {F}_k(S, U, \xi )$ .

Proposition 4.4. Fix $\xi \in E^0/{\sim } \ \text {and} \ v \in \xi $ . Let $\{S_e, U_g\}$ be a Cuntz–Krieger $(\mathcal {G}, E)$ -family. For $h \in \mathcal {G}^v_v$ , let $\overline {V}_h$ be as in Proposition 4.3. Then there is a homomorphism $\pi _{\overline {V}}: C^*(\mathcal {G}^v_v)\rightarrow \mathcal {M}C^*(S, U)$ that maps $\delta _h$ to $\overline {V}_h$ .

Proof. Let $h, k \in \mathcal {G}^v_v$ . Routine calculations show that for $k\geq 1$ and $\mu \in E^k\xi $ , we have $V_{h, \mu }V_{k, \mu }=V_{hk, \mu } $ and $V^*_{h, \mu } = S_{\mu } U_{g_{s(\,\mu )}}U_{h^{-1}}(S_{\mu } U_{g_{s(\,\mu )}})^* = V_{h^{-1}, \mu }. $ This implies that for any finite $F \subseteq E^k\xi $ ,

$$ \begin{align*} \sum_{\mu \in F} V_{h,\mu} \sum_{\mu \in F} V_{k, \mu}=\sum_{\mu \in F} V_{hk,\mu}. \end{align*} $$

Thus, $\overline {V}_h \overline {V}_k = \sum _{\mu \in E^k \xi }V_{h,\mu }\sum _{\mu \in E^k \xi }V_{k,\mu }=\overline {V}_{hk}$ and

$$ \begin{align*} \overline{V}_h^*=\sum_{\mu \in E^k \xi} V^*_{h,\mu}=\sum_{\mu \in E^k \xi}V_{h^{-1},\mu}=\overline{V}_{h^{-1}}. \end{align*} $$

So, the universal property of $C^*(\mathcal {G}^v_v)$ gives a homomorphism

$$ \begin{align*} \pi_{\overline{V}}: C^*(\mathcal{G}^v_v)\rightarrow \mathcal{M}(C^*(S, U)\quad \mathrm{such \ that} \ \pi_{\overline{V}}(\delta_h)=\overline{V}_h.\\[-34pt] \end{align*} $$

Proposition 4.5. Fix $\xi \in E^0/{\sim }$ . Let $\{S_e, U_g\}$ be a Cuntz–Krieger $(\mathcal {G}, E)$ -family. For $\mu , \beta \in E^k \xi $ , let $e_{\mu }\otimes e_{\beta }^*$ denote the rank-one operator on the Hilbert space $l^2(\{E^k \xi \})$ , and let $\Theta _{\mu , \beta }:=S_{\mu } U_{g_{s(\,\mu )}}U^*_{g_{s(\,\beta )}}S^*_{\beta } \in C^*(S, U)$ . Then there is an injective homomorphism

$$ \begin{align*}\theta: \mathcal{K}(l^2(\{E^k \xi\})) \rightarrow \overline{\mathrm{span}}\{\Theta_{\mu, \beta} \mid \mu, \beta \in E^k \xi\}\end{align*} $$

such that $\theta (e_{\mu }\otimes e_{\beta }^*)=\Theta _{\mu , \beta }$ .

Proof. We claim that the elements $\Theta _{\mu , \beta }$ are matrix units. Let $\mu , \beta , \alpha , \rho \in E^k \xi $ . Then,

$$ \begin{align*} \Theta_{\mu, \beta}\Theta_{\alpha, \rho}&= (S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta})(S_{\alpha} U_{g_{s(\alpha)}}U^*_{g_{s(\rho)}}S^*_{\rho})\\ &= \begin{cases} S_{\mu} U_{g_{s(\,\mu)}} U^*_{g_{s(\rho)}} S_{\rho}^* & \mathrm{if} \ \beta=\alpha, \\ 0 & \mathrm{otherwise,} \end{cases} \end{align*} $$

and $(\Theta _{\mu , \beta })^*=S_{\beta } U_{g_{s(\,\beta )}}U^*_{g_{s(\,\mu )}}S^*_{\mu }=\Theta _{\beta , \mu }$ . Hence, $\{\Theta _{\mu , \beta } \mid \mu , \beta \in E^k \xi \}$ is a family of matrix units. Since

$$ \begin{align*} \lVert S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta}\rVert^2 = \lVert S_{\beta} U_{g_{s(\,\beta)}}U^*_{g_{s(\,\mu)}}S^*_{\mu} S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta}\rVert =\lVert P_v\rVert = 1, \end{align*} $$

by Lemma 3.3, these are nonzero matrix units. Hence, by Corollary A.9 of [6], we get the injective homomorphism $\theta $ as claimed.

Proposition 4.6. Fix $\xi \in E^0/{\sim }$ and $v \in \xi $ . Let $\{S_e, U_g\}$ be a Cuntz–Krieger $(\mathcal {G}, E)$ -family. Let $\pi _{\overline {V}}$ and $\theta $ be as in Propositions 4.4 and 4.5, respectively. Then, there exists a homomorphism

$$ \begin{align*} \theta \otimes \pi_{\overline{V}}: \mathcal{K}(l^2(\{E^k \xi \})) \otimes C^*(\mathcal{G}^v_v) \rightarrow \mathcal{F}_k(S, U, \xi) \end{align*} $$

such that

$$ \begin{align*}\theta \otimes \pi_{\overline{V}}((e_{\mu}\otimes e_{\beta}^*) \otimes \delta_h)=\theta (e_{\mu}\otimes e_{\beta}^*) \pi_{\overline{V}}(\delta_h)=\pi_{\overline{V}}(\delta_h)\theta (e_{\mu}\otimes e_{\beta}^*)\end{align*} $$

for all $e_{\mu }\otimes e_{\beta }^* \in \mathcal {K}(l^2(\{E^k \xi \}))$ and for all $\delta _h \in C^*(\mathcal {G}^v_v)$ .

Proof. We have $\theta (e_{\mu }\otimes e_{\beta }^*)=\Theta _{\mu , \beta }$ and $\pi _{\overline {V}}(\delta _h)=\overline {V}_h$ in $\mathcal {F}_k(\xi )$ for all $e_{\mu }\otimes e_{\beta }^* \in \mathcal {K}(l^2(\{E^k \xi \}))$ and for all $\delta _h \in C^*(\mathcal {G}^v_v)$ . Then,

$$ \begin{align*} \Theta_{\mu, \beta}\overline{V}_h &= S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta} \overline{V}_h \\ &= S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta} \sum_{\gamma \in E^k \xi}S_\gamma U_{g_{s(\gamma)}}U_h U^*_{g_{s(\gamma)}}S^*_\gamma \\ &= \sum_{\gamma \in E^k \xi }S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta} S_\gamma U_{g_{s(\gamma)}}U_h U^*_{g_{s(\gamma)}}S^*_\gamma \\ &= S_{\mu} U_{g_{s(\,\mu)}}U_h U^*_{g_{s(\,\beta)}}S^*_{\beta}. \end{align*} $$

A similar calculation gives $\overline {V}_h \Theta _{\mu , \beta }=S_{\mu } U_{g_{s(\,\mu )}}U_h U^*_{g_{s(\,\beta )}}S^*_{\beta }$ . Hence, $\Theta _{\mu , \beta } \overline {V}_h=\overline {V}_h \Theta _{\mu , \beta }$ . We claim that

$$ \begin{align*} \overline{\mathrm{span}}\{\Theta_{\mu, \beta}\overline{V}_h \mid \mu, \beta \in E^k\xi, h \in \mathcal{G}^v_v\} =\mathcal{F}_k(S, U, \xi). \end{align*} $$

Let $\mu , \beta , \alpha , \rho \in E^k\xi $ and $h_1, h_2 \in \mathcal {G}^v_v$ . Then,

$$ \begin{align*} \Theta_{\mu, \beta}\overline{V}_{h_1} \Theta_{\alpha, \rho}\overline{V}_{h_2}=\Theta_{\mu, \beta}\Theta_{\alpha, \rho}\overline{V}_{h_1}\overline{V}_{h_2}= \delta_{\beta, \alpha}\Theta_{\mu, \rho}\overline{V}_{h_1h_2} \end{align*} $$

and $(\Theta _{\mu , \beta }\overline {V}_h)^*=\overline {V}_{h^{-1}}\Theta _{\beta , \mu }=\Theta _{\beta , \mu }\overline {V}_{h^{-1}}$ . So, $\overline {\mathrm { span}}\{\Theta _{\mu , \beta }\overline {V}_h \mid \mu , \beta \in E^k\xi , h \in \mathcal {G}^v_v\}$ is a $C^*$ -subalgebra of $\mathcal {F}_k(S, U, \xi )$ . Moreover, it contains the generators of $\mathcal {F}_k(S, U, \xi )$ , so it is all of $\mathcal {F}_k(S, U, \xi )$ .

Now the universal property of the (maximal) tensor product gives the desired homomorphism $\theta \otimes \pi _{\overline {V}}$ .

We show next the homomorphism $\theta \otimes \pi _{\overline {V}}$ is faithful. To show this, we need to verify that both $\theta $ and $\pi _{\overline {V}}$ are injective. From Proposition 4.5, we already know that $\theta $ is injective, so it suffices to show that $\pi _{\overline {V}}$ is injective as well.

Lemma 4.7. Fix $\xi \in E^0/{\sim }$ and $v \in \xi $ . Let $\{S_e, U_g\}$ be a Cuntz–Krieger $(\mathcal {G},E)$ -family. Suppose that the homomorphism $\pi _U: C^*(\mathcal {G}^v_v) \rightarrow C^*(S, U)$ that maps $\delta _h$ to $U_h$ is injective. Fix $k \in \mathbb {N}$ and $v\in E^0$ . Let $\overline {V}_h$ be as in Proposition 4.3. Then, the homomorphism $\pi _{\overline {V}}^{(v,k)}: C^*(\mathcal {G}^v_v) \rightarrow \mathcal {F}_k(S, U, \xi )$ that maps $\delta _h$ to $\overline {V}_h$ is injective.

Proof. Fix $\lambda \in E^k \xi $ and let $Y_{\lambda }=S_{\lambda } U_{g_{s(\lambda )}}$ . Then,

$$ \begin{align*}Y^*_{\lambda} \overline{V}_h Y_{\lambda} = \sum_{\mu \in E^k \xi} U^*_{g_{s(\lambda)}}S^*_{\lambda} S_{\mu} U_{g_{s(\,\mu)}}U_h U^*_{g_{s(\,\mu)}}S^*_{\mu} S_{\lambda} U_{g_{s(\lambda)}} = U_h. \end{align*} $$

Define $\mathrm {Ad}_{Y_{\lambda }}: \mathcal {F}_k(S, U, \xi ) \rightarrow C^*(S, U)$ by $\mathrm {Ad}_{Y_{\lambda }}(a)=Y^*_{\lambda } a Y_{\lambda }$ . By linearity and continuity, $\mathrm {Ad}_{Y_{\lambda }} \circ \pi _{\overline {V}}^{(v,k)}=\pi _U$ . Hence, $\mathrm {Ad}_{Y_{\lambda }} \circ \pi _{\overline {V}}^{(v,k)}$ is injective, so $\pi _{\overline {V}}^{(v,k)}$ is also injective.

Since $\mathcal {K}(l^2(\{E^k \xi \}))$ is simple and nuclear, Proposition 4.5 and Lemma 4.7 show that if $\pi _U$ is injective on $C^*(\mathcal {G}^v_v)$ , then the homomorphism of Proposition 4.6 is an isomorphism. So,

(4.2) $$ \begin{align} \mathcal{F}_k(\xi)\cong \mathcal{K}(l^2(\{E^k \xi \}))\otimes C^*(\mathcal{G}^v_v). \end{align} $$

Moreover, we obtain the following corollary. Recall that

$$ \begin{align*} \mathcal{F}_k=\overline{\mathrm{span}}\{s_{\mu} u_g s^*_{\beta} \mid s(\,\mu)= g \cdot s(\,\beta), \,\mathrm{and}\, \lvert \,\mu \rvert=\lvert \,\beta \rvert=k \}. \end{align*} $$

Corollary 4.8. Let $(\mathcal {G},E)$ be a self-similar groupoid. Fix $\xi \in E^0/{\sim }$ and $v \in \xi $ . Suppose that for each $k \in \mathbb {N}$ , there exists $\lambda \in vE^k$ such that the map $g\mapsto (g\cdot \lambda )\varphi (g,\lambda )$ is injective. Then,

$$ \begin{align*} \mathcal{F}_k \cong \bigoplus_{\xi \in E^0/\sim}\mathcal{F}_k(\xi)\cong\bigoplus_{\xi \in E^0/\sim}\mathcal{K}(l^2(\{E^k \xi \}))\otimes C^*(\mathcal{G}^v_v). \end{align*} $$

Proof. For $\mu , \beta , \alpha , \rho \in E^k$ with $s(\,\mu )=g \cdot s(\,\beta ) \in \xi _1$ and $s(\alpha )= h \cdot s(\rho ) \in \xi _2$ , the equation of Lemma 3.3 gives

$$ \begin{align*}(s_{\mu} u_g s^*_{\beta})(s_{\alpha} u_h s^*_{\rho})=\begin{cases} s_{\mu} u_{g}u_{h} s^{*}_{\rho} & \mathrm{if} \ \beta=\alpha, \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$

Hence, $\mathcal {F}_k(\xi _1) \mathcal {F}_k(\xi _2)=0$ , when $\xi _1 \neq \xi _2$ , so Corollary A.11 of [6] combined with (4.2) gives an isomorphism of $\bigoplus _{\xi \in E^0/\sim } \mathcal {F}_k(\xi )$ onto $\mathcal {F}_k$ . Equation (4.2) gives the second isomorphism.

Corollary 4.9. Let $(\mathcal {G},E)$ be a self-similar groupoid. Then,

$$ \begin{align*}C^*(\mathcal{G}, E)^{\gamma} =\overline{\bigcup_k \mathcal{F}_k}=\overline{\bigcup_k \bigg(\bigoplus_{\xi \in E^0/\sim}\mathcal{F}_k(\xi)\bigg)}.\end{align*} $$

Proof. For any k, we claim that $\mathcal {F}_k \subset \mathcal {F}_{k+1}$ . Fix $\mu , \beta \in E^k,g \in \mathcal {G}$ with $s(\,\mu )=g \cdot s(\,\beta )$ . We have

$$ \begin{align*}s_{\mu} u_g s^*_{\beta} = s_{\mu} u_g u_{s(g)}s^*_{\beta} = \sum_{e \in s(g)E^1}s_{\mu} u_g s_e s^*_e s^*_{\beta} = \sum_{e \in s(g)E^1}s_{\mu (g\cdot e)}u_{\varphi(g,e)}s^*_{\beta e} \in \mathcal{F}_{k+1}.\end{align*} $$

Hence, $\mathcal {F}_k \subset \mathcal {F}_{k+1}$ for all k. By Corollary 4.1, the claim follows.

Lemma 4.10. Let $(\mathcal {G}, E)$ be a self-similar groupoid. Suppose that $\{T_e,W_g\}$ is a $(\mathcal {G},E)$ -family in a $C^*$ -algebra B. Let

$$ \begin{align*}\pi_{T,W} : C^*(\mathcal{G},E)\rightarrow C^*(T,W)\end{align*} $$

be the homomorphism induced by the universal property. Suppose that for each $v \in E^0$ , the homomorphism $\pi _{v,W}: C^*(\mathcal {G}^v_v) \rightarrow C^*(T,W)$ such that $\pi _{v,W}(\delta _g)=W_g$ for all g is injective. Then, $\pi _{T,W}$ is isometric on $C^*(\mathcal {G},E)^{\gamma }$ .

Proof. Fix $\xi \in E^0/{\sim }$ and $ v \in \xi $ . Choose elements $g_w\in \mathcal {G}^w_v$ for $w\in \xi $ with $g_v=v$ . For $h \in \mathcal {G}^v_v$ and $k\in \mathbb {N}$ , let $\overline {W}_h=\sum _{\mu \in E^k \xi } T_{\mu } W_{g_{s(\,\mu )}}W_h W^*_{g_{s(\,\mu )}}T^*_{\mu }$ as in Proposition 4.3. Lemma 4.7 shows that the homomorphism $\pi _{\overline {W}}: C^*(\mathcal {G}^v_v) \rightarrow \mathcal {M}C^*(T,W)$ is injective.

Let $\theta \otimes \pi _{\overline {W}}$ be as in Proposition 4.6. Since $\mathcal {K}(l^2(E^k\xi ))$ is simple and nuclear, and since each $T_{\mu } T^*_{\beta } \neq 0$ , the map $\pi _{T,W} \circ (\theta \otimes \pi _{\overline {u}})=\theta \otimes \pi _{\overline {W}}$ is injective on each $\mathcal {F}_k(\xi )$ . Therefore, it is also injective on $\mathcal {F}_{k}=\bigoplus _{\xi \in E^0/\sim }\mathcal {F}_{k}(\xi )$ . Because every injective $C^*$ -algebra homomorphism is isometric, $\pi _{T,W}$ is isometric on $\mathcal {F}_{k}$ . Hence, $\pi _{T, W}$ is isometric on $\bigcup _k\mathcal {F}_{k}$ and hence on $\overline {U_k\mathcal {F}_k}=C^*(\mathcal {G},E)^{\gamma }$ .

5 The gauge-invariant uniqueness theorem

Theorem 5.1. Let $(\mathcal {G}, E)$ be a self-similar groupoid. Suppose that $(T,W)$ is a $(\mathcal {G}, E)$ -family in a $C^*$ -algebra B. The universal property of $C^*(\mathcal {G}, E)$ gives a homomorphism

$$ \begin{align*} \pi_{T,W} : C^*(\mathcal{G},E)\rightarrow C^*(T,W). \end{align*} $$

If there is a continuous action $\eta : \mathbb {T} \rightarrow \mathrm {Aut} B$ such that $\eta _z(T_e)=zT_e$ and $\eta _z(W_g)=W_g$ for all $e \in E^1$ and $g \in G$ , and if the homomorphism $\pi _{v,W}$ is injective for each $v \in E^0$ , then $\pi _{T,W}$ is an isomorphism of $C^*(\mathcal {G},E)$ onto $C^*(T,W)$ .

Proof. Let $\Phi : C^*(\mathcal {G}, E) \rightarrow C^*(\mathcal {G}, E)^{\gamma }$ be the faithful conditional expectation of Corollary 4.1. Let $\Psi :C^*(T,W)\rightarrow C^*(T,W)^{\eta }$ be the corresponding expectation obtained from $\eta $ . Since $\eta _z \circ \pi _{T,W}$ and $\pi _{T,W} \circ \gamma _z$ agree on generators, they are equal. Hence, $\Psi \circ \pi _{T,W} =\pi _{T,W} \circ \Phi $ . By [8, Lemma 3.14], $\pi _{T,W}$ is injective if it is injective on $C^*(\mathcal {G}, E)^{\gamma }$ , which it is by Lemma 4.10.