TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

Abstract

Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists.

We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

1 Introduction

Groupoid $C^*$ -algebras have played an important role in the subject of $C^*$ -algebras since Renault’s seminal monograph [32]. An important development was a series of works, principally by Renault [32, 33] and Kumjian [19], in which it was shown that Cartan pairs of $C^*$ -algebras correspond to twisted groupoid $C^*$ -algebras for a certain class of étale groupoids. This was used to good effect by Matui and Matsumoto in the study of symbolic dynamics via Cuntz–Krieger algebras [23].

An étale groupoid whose unit space is Hausdorff and has a basis of compact open sets is said to be ample [31]. The author introduced in [36] algebras over any base commutative ring associated to ample groupoids, nowadays called ‘Steinberg algebras’, which are ring-theoretic analogues of groupoid $C^*$ -algebras. They include group algebras, inverse semigroup algebras and Leavitt path algebras [1, 35]. In recent years they have been the subject of investigation by a number of authors. See, for example, [3, 7, 10, 13, 14, 24–30, 35–42].

There were, in particular, a number of papers investigating when an ample groupoid can be recovered from the pair of its Steinberg algebra and the ‘diagonal’ subalgebra of functions on the unit space; see [2, 11, 13, 39]. This is similar in spirit to the work of Kumjian and Renault and may have motivated the authors of [5] to introduce twisted Steinberg algebras of ample groupoids. A theory of algebraic Cartan pairs was subsequently developed in [4], closely paralleling the Renault–Kumjian theory from the $C^*$ -algebra setting.

This paper arose from an attempt by the author to understand [4] from the point of view of inverse semigroup theory. It has been known for a long time that there is a close connection between ample groupoids and inverse semigroups [16, 17, 22, 31, 32]. The strategy used by the author to reconstruct an ample groupoid from its Steinberg algebra and diagonal subalgebra in [39] was as follows. From a Steinberg algebra of an ample groupoid ${\mathscr G}$ and its diagonal subalgebra, one obtains an exact sequence of inverse semigroups $K\to S\to S/K$ , where S is the normalizer of the diagonal subalgebra and K is the semigroup of diagonal normalizers (which is a normal inverse subsemigroup of S). It turns out that, if the original groupoid is effective, for example, then $S/K$ is isomorphic to the inverse semigroup of compact open bisections of ${\mathscr G}$ . Since ${\mathscr G}$ can be reconstructed from its inverse semigroup of compact open bisections via Exel’s ultrafilter groupoid construction [16, 17], this shows that ${\mathscr G}$ is determined by its Steinberg algebra and diagonal subalgebra.

Therefore, it is natural to attempt to show that discrete twists over ample groupoids correspond to certain exact sequences of inverse semigroups. Then one could re-prove some of the results of [4] via purely inverse-semigroup-theoretic means by showing that the exact sequence corresponding to the twist is equivalent to the exact sequence coming from the Cartan pair. This paper sets the foundations for such an approach.

The papers of Bice [8, 9] seem to be related to ours in that they attempt to understand related algebras using inverse semigroup theory, but they apply neither the categorical and cohomological approach we do, nor the crossed product construction.

We begin by giving a covariant equivalence of categories between ample groupoids and appropriately restricted functors, and Boolean inverse semigroups and appropriately restricted homomorphisms. Moreover, we show that these functors send exact sequences of ample groupoids to exact sequences of inverse semigroups, and vice versa. We then describe the exact sequences of inverse semigroups corresponding to twists over an ample groupoid by a discrete abelian group A (written multiplicatively). We show that A-twists over ${\mathscr G}$ correspond to extensions of the inverse semigroup $\Gamma _c({\mathscr G})$ of compact open bisections of ${\mathscr G}$ by the commutative inverse semigroup $\widetilde {A}$ of compactly supported locally constant functions $f\colon {\mathscr G}^{(0)}\to A\cup \{0\}$ . Such extensions are then classified by Lausch’s cohomology theory of inverse semigroups [20]. In particular, we show that the abelian group of equivalence classes of A-twists under Baer sum is isomorphic to Lausch’s second cohomology group $H^2(\Gamma _c({\mathscr G}),\widetilde {A})$ . This is similar in spirit to how Lausch cohomology arose in the work on Cartan pairs of von Neumann algebras in [15]. We also show that the question of whether the twist admits a continuous global section can be interpreted inverse-semigroup-theoretically using ideas of [15].

In the final section we introduce what we believe to be a new notion of inverse semigroup crossed product, arising from an action of an inverse semigroup on a ring and a $2$ -cocycle with respect to the action (a related, but different $C^*$ -algebraic notion is in [12]). It generalizes the skew inverse semigroup ring construction for a large class of actions (those in [18]), as well as group crossed products (with respect to an action). After developing the basic properties of the crossed product (including a universal property), we show that twisted Steinberg algebras are inverse semigroup crossed products with respect to the same Lausch $2$ -cocycle that classifies the twist under our bijection between twists and cohomology classes.

2 Groupoids and inverse semigroups

Here we recall some basic notions about inverse semigroups and groupoids.

2.1 Inverse semigroups

An inverse semigroup is a semigroup S such that, for each $s\in S$ , there is a unique element $s^*\in S$ that satisfies $ss^*s=s$ and $s^*ss^*=s^*$ . In an inverse semigroup, the idempotents commute and, hence, form a subsemigroup $E(S)$ . Moreover, $e^*=e$ for any idempotent e. Note that $ss^*,s^*s\in E(S)$ for any $s\in S$ , as is $ses^*$ for any $e\in E(S)$ . We also observe that $(st)^*=t^*s^*$ . An element s of a semigroup S is (von Neumann) regular if $s=ss's$ for some $s'\in S$ ; a semigroup S is (von Neumann) regular if all its elements are regular. It is well known that inverse semigroups are precisely the regular semigroups with commuting idempotents.

There is a natural partial order on S given by $s\leq t$ if $s=te$ for some idempotent $e\in E(S)$ or, equivalently, $s=ft$ for some $f\in E(S)$ . One can, in fact, take $e=s^*s$ and $f=ss^*$ . The natural partial order is compatible with multiplication and preserved (not reversed) by the involution. If $e,f\in E(S)$ , then $ef$ is the meet of $e,f$ in the natural partial order and so $E(S)$ is a meet semilattice. Any homomorphism ${\varphi }\colon S\to T$ of inverse semigroups automatically preserves the involution and order. We say that ${\varphi }$ is idempotent separating if ${\varphi }|_{E(S)}$ is injective.

A normal inverse subsemigroup of an inverse semigroup S is an inverse subsemigroup K with $E(K)=E(S)$ and $sKs^*\subseteq K$ for all $s\in S$ . If ${\varphi }\colon S\to T$ is a homomorphism of inverse semigroups, then $\ker {\varphi }={\varphi ^{-1}}(E(T))$ is a normal inverse subsemigroup and idempotent separating homomorphisms are ‘determined’ by their kernels. See [21] for basics on inverse semigroup theory.

Many inverse semigroups have a zero element. If S is an inverse semigroup with zero, then we say that $s,t\in S$ are orthogonal if $st^*=0=t^*s$ . A Boolean inverse semigroup is an inverse semigroup S with zero such that $E(S)$ is a Boolean algebra, and which admits joins of orthogonal pairs of elements. When we say that $E(S)$ is a Boolean algebra, we mean that it admits finite joins, these joins distribute over meets, and it has relative complements (if $f\leq e$ , then there exists $e\setminus f$ with $f(e\setminus f)=0$ and $e=f\vee (e\setminus f)$ ). Note that, if $s,t$ are orthogonal, then we have $(s\vee t)(s\vee t)^* = ss^*\vee tt^*$ and $(s\vee t)^*(s\vee t)=s^*s\vee t^*t$ . A homomorphism ${\varphi }\colon S\to T$ of Boolean inverse semigroups is called additive if it preserves joins of orthogonal idempotents, in which case it preserves all finite joins existing in S. There are a number of other axiomatizations of Boolean inverse semigroups, and we refer the reader to [43] for more details.

If X is a topological space, then the set $I_X$ of all homeomorphisms between open subsets of X is an inverse semigroup under composition of partial functions. An action of an inverse semigroup S on a space X by partial homeomorphisms is just a homomorphism $\alpha \colon S\to I_X$ . The action is nondegenerate if X is the union of the domains of the elements of $\alpha (S)$ .

2.2 Groupoids

A groupoid ${\mathscr G}$ is a small category in which each arrow is invertible. We take the approach here, popular in analysis, of viewing ${\mathscr G}$ as a set with a partially defined multiplication and a totally defined inversion. The objects of ${\mathscr G}$ are identified with identity arrows (also called units) and the unit space is denoted by ${\mathscr G}^{(0)}$ . We use ${\mathop {\boldsymbol d}}\colon {\mathscr G}\to {\mathscr G}^{(0)}$ and ${\mathop {\boldsymbol r}}\colon {\mathscr G}\to {\mathscr G}^{(0)}$ for the domain and range maps, respectively.

A topological groupoid is a groupoid endowed with a topology that makes the multiplication and inversion maps continuous. As ${\mathop {\boldsymbol d}}(g)=g^{-1} g$ and ${\mathop {\boldsymbol r}}(g)=gg^{-1}$ , the domain and range maps are also continuous. Here we give ${\mathscr G}^{(0)}$ the subspace topology. When working with topological groupoids, we want our functors to be continuous.

A topological groupoid ${\mathscr G}$ is étale if the domain map is a local homeomorphism. This is equivalent to the range map being a local homeomorphism and also to the multiplication map being a local homeomorphism. The unit space ${\mathscr G}^{(0)}$ of an étale groupoid is open. See [34] for details. A (local) bisection of an étale groupoid is an open subset $U\subseteq {\mathscr G}$ such that ${\mathop {\boldsymbol d}}|_U$ and ${\mathop {\boldsymbol r}}|_U$ are injective; some authors do not require bisections to be open and so we often say ‘open bisection’ for emphasis. The origin of the term ‘bisection’ is that they correspond to subsets of ${\mathscr G}$ that are simultaneously the image of a local section of ${\mathop {\boldsymbol d}}$ and of ${\mathop {\boldsymbol r}}$ . If $U,V$ are bisections, then so are $UV=\{gg'\mid g\in U, g'\in V\}$ and $U^*=\{g^{-1}\mid g\in U\}$ . The bisections form an inverse semigroup with respect to this product operation, taking $U^*$ as the inverse of U. The idempotent bisections are the open subsets of ${\mathscr G}^{(0)}$ and the natural partial order is just containment.

An étale groupoid ${\mathscr G}$ is called ample (following Paterson [31]) if ${\mathscr G}^{(0)}$ is Hausdorff and has a basis of compact open sets. In this case, the compact open bisections form a basis for the topology of ${\mathscr G}$ . The collection $\Gamma _c({\mathscr G})$ of compact open bisections of ${\mathscr G}$ is a Boolean inverse semigroup. References for ample groupoids include [16, 31, 36]. Ample groupoids arise in nature as groupoids of germs of actions of inverse semigroups on Hausdorff spaces with bases of compact open sets.

3 Extensions of groupoids and inverse semigroups

3.1 An equivalence of categories

In this subsection we show that the bijection between isomorphism classes of ample groupoids and Boolean inverse semigroups arising from the work of Exel [16, 17] and Lawson and Lenz [22] can be turned into a covariant equivalence of categories if we restrict our morphisms suitably. We use this equivalence to show that discrete twists over an ample groupoid correspond to certain idempotent-separating extensions of inverse semigroups that are classified by Lausch cohomology [20].

We call a continuous functor ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ of topological groupoids iso-unital if ${\varphi }|_{{\mathscr G}^{(0)}}\colon {\mathscr G}^{(0)}\to {\mathscr H}^{(0)}$ is a homeomorphism. Note that, if ${\mathscr G}$ and ${\mathscr H}$ are étale and ${\varphi }$ is open, then this is equivalent to ${\varphi }|_{{\mathscr G}^{(0)}}\colon {\mathscr G}^{(0)}\to {\mathscr H}^{(0)}$ being bijective. Notice that if ${\varphi }$ is iso-unital, then ${\mathop {\boldsymbol d}}({\varphi }(g)) = {\mathop {\boldsymbol d}}({\varphi }(g'))$ if and only if ${\mathop {\boldsymbol d}}(g)={\mathop {\boldsymbol d}}(g')$ , and, dually, that ${\mathop {\boldsymbol r}}({\varphi }(g))={\mathop {\boldsymbol r}}({\varphi }(g'))$ if and only if ${\mathop {\boldsymbol r}}(g)={\mathop {\boldsymbol r}}(g')$ .

Proposition 3.1. Let ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ be an iso-unital functor between étale groupoids. Then the restriction of ${\varphi }$ to any bisection is injective. Consequently, ${\varphi }$ is open if and only if it is a local homeomorphism.

Proof. Let $U\subseteq {\mathscr G}$ be a bisection. If $g_1,g_2\in U$ satisfy ${\varphi }(g_1)={\varphi }(g_2)$ , then, since ${\varphi }$ is iso-unital, we must have ${\mathop {\boldsymbol d}}(g_1)={\mathop {\boldsymbol d}}(g_2)$ ; and so $g_1=g_2$ as U is a bisection. Thus ${\varphi }|_U$ is injective. Clearly, if ${\varphi }$ is a local homeomorphism, it is open. Conversely, if ${\varphi }$ is open and $g\in {\mathscr G}$ , then, since ${\mathscr G}$ is étale, we can find an open bisection U with $g\in U$ . Then ${\varphi }|_U\colon U\to {\varphi }(U)$ is injective and open and, hence, a homeomorphism. Therefore, ${\varphi }$ is a local homeomorphism.

The corresponding notion for inverse semigroups is the following. A homomorphism ${\varphi }\colon S\to T$ of inverse semigroups is idempotent bijective if ${\varphi }|_{E(S)}\colon E(S)\to E(T)$ is a bijection. Note that, if S and T are Boolean inverse semigroups, then any idempotent bijective homomorphism is additive.

The following lemma combines two well-known facts, but we include its proof for completeness.

Lemma 3.2. Let ${\varphi }\colon S\to T$ be an idempotent bijective homomorphism of inverse semigroups and $\psi \colon {\mathscr G}\to {\mathscr H}$ an iso-unital functor between topological groupoids. Then

1. (1) ${\varphi }$ is injective if and only if $\ker {\varphi }={\varphi }^{-1} (E(T))=E(S)$ ;

2. (2) $\psi $ is injective if and only if $\psi ^{-1}({\mathscr H}^{(0)})={\mathscr G}^{(0)}$ .

Proof. If ${\varphi }$ is injective, then ${\varphi }(s)=e\in E(T)$ implies that ${\varphi }(s^*s)=e^*e=e$ and so $s=s^*s\in E(S)$ . Conversely, if ${\varphi }^{-1}(E(T))=E(S)$ and ${\varphi }(s_1)={\varphi }(s_2)$ , then since ${\varphi }$ is idempotent bijective, $s_1^*s_1=s_2^*s_2$ . Also ${\varphi }(s_1s_2^*)={\varphi }(s_1s_1^*)\in E(T)$ and so $s_1s_2^*\in E(S)$ . Therefore, $s_1=s_1s_1^*s_1=s_1s_2^*s_2\leq s_2$ ; dually, $s_2\leq s_1$ and, hence, $s_1=s_2$ . Thus ${\varphi }$ is injective.

Similarly, if $\psi $ is injective, then $\psi (g)=x\in {\mathscr H}^{(0)}$ implies that $\psi (g)=\psi ({\mathop {\boldsymbol d}}(g))$ , and so $g={\mathop {\boldsymbol d}}(g)\in {\mathscr G}^{(0)}$ . On the other hand, if $\psi ^{-1}({\mathscr H}^{(0)})={\mathscr G}^{(0)}$ and $\psi (g_1)=\psi (g_2)$ , then ${\mathop {\boldsymbol d}}(g_1)={\mathop {\boldsymbol d}}(g_2)$ and ${\mathop {\boldsymbol r}}(g_1)={\mathop {\boldsymbol r}}(g_2)$ , as $\psi $ is iso-unital; moreover, $\psi (g_1g_2^{-1})=\psi (g_1g_1^{-1})\in {\mathscr H}^{(0)}$ , and so $g_1g_2^{-1} \in {\mathscr G}^{(0)}$ , whence $g_1=g_2$ .

To every ample groupoid ${\mathscr G}$ we can associate the Boolean inverse semigroup $\Gamma _c({\mathscr G})$ . We proceed with the inverse construction. If S is a Boolean inverse semigroup, then ${\mathop {\mathrm {Spec}}}(E(S))$ denotes the Stone space of the Boolean algebra $E(S)$ . Its elements are the nonzero Boolean algebra homomorphisms (characters) $\chi \colon E\to \{0,1\}$ . A basis of compact open subsets for ${\mathop {\mathrm {Spec}}}(E(S))$ consists of the sets of the form

$$ \begin{align*} D(e) = \{\chi\in {\mathop{\mathrm{Spec}}}(E(S))\mid \chi(e)=1\}. \end{align*} $$

In fact $e\mapsto D(e)$ is an isomorphism of Boolean algebras between $E(S)$ and the Boolean algebra of compact open subsets of ${\mathop {\mathrm {Spec}}}(E(S))$ . There is a natural action of S on ${\mathop {\mathrm {Spec}}}(E(S))$ by partial homeomorphisms. For each $s\in S$ , there is a homeomorphism $\beta _s\colon D(s^*s)\to D(ss^*)$ given by $\beta _s(\chi )(e) = \chi (s^*es)$ , and the assignment $s\mapsto \beta _s$ is a nondegenerate action of S by partial homeomorphisms. We often write $s\chi $ instead of $\beta _s(\chi )$ . Put ${\mathcal G}(S)=S\ltimes {\mathop {\mathrm {Spec}}}(E(S))$ , the groupoid of germs. Recall that

$$ \begin{align*} {\mathcal G}(S) = (S\times {\mathop{\mathrm{Spec}}}(B))/{\sim}, \end{align*} $$

where $(s,\chi )\sim (t,\lambda )$ if and only if $\chi =\lambda $ and there exists $u\leq s,t$ with $\chi (u^*u)=1$ . We write $[s,\chi ]$ for the equivalence class of $(s,\chi )$ . The groupoid multiplication $[s,\chi ][t,\lambda ]$ is defined by if and only if $\chi = t\lambda $ , in which case the product is $[st,\lambda ]$ . A basis for the topology is given by the compact open bisections

$$ \begin{align*} D(s) = \{[s,\chi]\mid \chi\in D(s^*s)\}. \end{align*} $$

This topology makes ${\mathcal G}(S)$ into an ample groupoid. The unit space ${\mathcal G}(S)^{(0)}$ can be identified homeomorphically with ${\mathop {\mathrm {Spec}}}(E(S))$ via $\chi \mapsto [e,\chi ]$ , where $e\in E(S)$ with $\chi (e)=1$ . Under this identification, ${\mathop {\boldsymbol d}}([s,\chi ])=\chi $ and ${\mathop {\boldsymbol r}}([s,\chi ]) = s\chi $ . Inversion is given by $[s,\chi ]^{-1}=[s^*,s\chi ]$ . See [16, 31, 36] for more on groupoids of germs. (In [22], the authors work with a groupoid of ultrafilters on the poset S, but it is well known and easy to see that this groupoid is isomorphic to ${\mathcal G}(S)$ .)

The following theorem combines work of Exel [16, 17] and of Lawson and Lenz [22].

Theorem 3.3 (Exel, Lawson and Lenz)

Let ${\mathscr H}$ be an ample groupoid and S a Boolean inverse semigroup. Then

1. (1) there is an isomorphism $\eta _{{\mathscr H}}\colon {\mathscr H}\to {\mathcal G}(\Gamma _c({\mathscr H}))$ given by $\eta _{{\mathscr H}}(h) = [U,\chi _{{\mathop {\boldsymbol d}}(h)}]$ , where U is any compact open bisection containing h and $\chi _x$ is the characteristic function for the set of compact open subsets of ${\mathscr H}^{(0)}$ containing $x\in {\mathscr H}^{(0)}$ ;

2. (2) there is an isomorphism $\varepsilon _S\colon S\to \Gamma _c({\mathcal G}(S))$ given by $\varepsilon _S(s) = D(s)$ .

We now show that the constructions ${\mathscr H}\mapsto \Gamma _c({\mathscr H})$ and $S\mapsto {\mathcal G}(S)$ can be promoted to inverse equivalences between the category of ample groupoids with open iso-unital functors and the category of Boolean inverse semigroups with idempotent bijective homomorphisms.

Theorem 3.4. The category of ample groupoids with open iso-unital functors is equivalent to the category of Boolean inverse semigroups with idempotent bijective homomorphisms. More precisely, $\Gamma _c$ is a functor from the category of ample groupoids with open iso-unital functors to the category of Boolean inverse semigroups with idempotent bijective homomorphisms, where, if ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ is an open iso-unital functor, then $\Gamma _c({\varphi })\colon \Gamma _c({\mathscr G})\to \Gamma _c({\mathscr H})$ is given by $\Gamma _c({\varphi })(U)={\varphi }(U)$ . Moreover, the groupoid of germs construction $S\mapsto {\mathcal G}(S)$ provides a quasi-inverse functor such that, if ${\varphi }\colon S\to T$ is idempotent bijective, then ${\mathcal G}({\varphi })([s,\chi ]) = [{\varphi }(s), \chi \circ ({\varphi }|_{E(S)})^{-1}]$ .

Proof. We begin by showing that $\Gamma _c$ is a functor. Let ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ be an open iso-unital functor. Let $U\in \Gamma _c({\mathscr G})$ . Then ${\varphi }(U)$ is compact open; we just need to show that it is a bisection. Let $g_1,g_2\in U$ satisfy ${\mathop {\boldsymbol d}}({\varphi }(g_1))={\mathop {\boldsymbol d}}({\varphi }(g_2))$ . Then ${\mathop {\boldsymbol d}}(g_1)={\mathop {\boldsymbol d}}(g_2)$ as ${\varphi }$ is iso-unital, and $g_1=g_2$ because U is a bisection. Thus ${\mathop {\boldsymbol d}}|_{{\varphi }(U)}$ is injective and the same argument shows that ${\mathop {\boldsymbol r}}|_{{\varphi }(U)}$ is injective. We conclude that ${\varphi }(U)\in \Gamma _c({\mathscr G})$ .

Next we check that ${\varphi }(UV)={\varphi }(U){\varphi }(V)$ for $U,V\in \Gamma _c({\mathscr G})$ . If $g\in UV$ , then $g=g_1g_2$ for some $g_1\in U$ and $g_2\in V$ , and so ${\varphi }(g)={\varphi }(g_1){\varphi }(g_2)\in {\varphi }(U){\varphi }(V)$ . Thus ${\varphi }(UV)\subseteq {\varphi }(U){\varphi }(V)$ . Conversely, suppose that $h\in {\varphi }(U){\varphi }(V)$ . Then $h={\varphi }(g_1){\varphi }(g_2)$ for some $g_1\in U$ and $g_2\in V$ . Since ${\varphi }$ is iso-unital, this implies that ${\mathop {\boldsymbol d}}(g_1)={\mathop {\boldsymbol r}}(g_2)$ and so $h={\varphi }(g_1g_2)$ for some $g_1g_2\in UV$ . Thus ${\varphi }(UV)={\varphi }(U){\varphi }(V)$ as required. Observe that, since ${\varphi }|_{{\mathscr G}^{(0)}}\colon {\mathscr G}^{(0)}\to {\mathscr H}^{(0)}$ is a homeomorphism and $E(\Gamma _c({\mathscr G}))$ is the set of compact open subsets of ${\mathscr G}^{(0)}$ and $E(\Gamma _c({\mathscr H}))$ is the set of compact open subsets of ${\mathscr H}^{(0)}$ , it follows that $\Gamma _c({\varphi })$ is idempotent bijective. It is clear from the definition that $\Gamma _c$ is functorial.

Next we show that, if ${\varphi }\colon S\to T$ is an idempotent bijective homomorphism of Boolean inverse semigroups, then ${\mathcal G}({\varphi })$ is a well-defined, open iso-unital functor. To simplify our notation, we write ${\varphi }_*$ for ${\mathcal G}({\varphi })$ . To check that ${\varphi }_*$ is well defined, suppose that $u\leq s,s'$ satisfies $\chi (u^*u)=1$ . Then ${\varphi }(u)\leq {\varphi }(s),{\varphi }(s')$ and we have that $\chi \circ ({\varphi }_{E(S)})^{-1}({\varphi }(u)^*{\varphi }(u)) = \chi (u^*u)=1$ . Thus $[{\varphi }(s),\chi \circ ({\varphi }_{E(S)})^{-1}]=[{\varphi }(s'),\chi \circ ({\varphi }_{E(S)})^{-1}]$ . It is immediate from ${\varphi }$ being a homomorphism that ${\varphi }_*$ is a functor. It is bijective on unit spaces as $\chi \mapsto \chi \circ ({\varphi }_{E(S)})^{-1}$ is a bijection of Stone spaces. It remains to show that ${\varphi }_*$ is open and continuous. For continuity, we claim that ${\varphi }_*^{-1}(D(t)) = \bigcup _{{\varphi }(s)\leq t} D(s)$ . Indeed, if ${\varphi }(s)\leq t$ and $\chi (s^*s)=1$ , then ${\varphi }_*([s,\chi ]) = [{\varphi }(s),\chi \circ ({\varphi }_{E(S)})^{-1}] = [t,\chi \circ ({\varphi }|_{E(S)})^{-1}]\in D(t)$ . Conversely, if ${\varphi }_\ast ([s,\chi ])=[{\varphi }(s),\chi \circ ({\varphi }_{E(S)})^{-1}]\in D(t)$ , then there exists $u\leq {\varphi }(s),t$ with $\chi \circ ({\varphi }_{E(S)})^{-1}(u^*u)=1$ . Put $e=({\varphi }_{E(S)})^{-1}(u^*u)\leq ({\varphi }_{E(S)})^{-1}({\varphi }(s)^*{\varphi }(s))=s^*s$ and let $s_0=se$ . Then ${\varphi }(s_0) = {\varphi }(s){\varphi }(e)= {\varphi }(s)u^*u=u\leq t$ and $\chi (s_0^*s_0) = \chi (e) = \chi \circ ({\varphi }_{E(S)})^{-1}(u^*u)=1$ . Thus $[s,\chi ] = [s_0,\chi ]\in D(s_0)$ satisfies ${\varphi }(s_0)\leq t$ . To show that ${\varphi }_*$ is open, we claim that ${\varphi }_*(D(s)) = D({\varphi }(s))$ . From what we have just shown, ${\varphi }_*(D(s))\subseteq D({\varphi }(s))$ . Suppose that $[{\varphi }(s),\lambda ]\in D({\varphi }(s))$ . Let $\chi =\lambda \circ {\varphi }|_{E(S)}$ . Then $\chi (s^*s) = \lambda ({\varphi }(s^*s)) = 1$ and so $[s,\chi ]\in D(s)$ . Moreover, ${\varphi }_*([s,\chi ]) = [{\varphi }(s), \chi \circ ({\varphi }|_{E(S)})^{-1}] = [{\varphi }(s),\lambda ]$ , as required. It is clear that if ${\varphi },\psi $ are composable idempotent bijective homomorphisms, then ${\mathcal G}({\varphi }\psi ) = {\mathcal G}({\varphi })\circ {\mathcal G}(\psi )$ .

It now remains to show that $\eta $ and $\varepsilon $ from Theorem 3.3 are natural isomorphisms. Let ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ be an open iso-unital functor. Let $g\in {\mathscr G}$ and U be a compact open bisection containing g. Note that ${\varphi }(U)$ is a compact open bisection containing ${\varphi }(g)$ and ${\mathcal G}(\Gamma _c({\varphi }))(\eta _{{\mathscr G}}(g)) = {\mathcal G}(\Gamma _c({\varphi }))[U,\chi _{{\mathop {\boldsymbol d}}(g)}] = [{\varphi }(U),\chi _{{\mathop {\boldsymbol d}}({\varphi }(g))}] = \eta _{{\mathscr H}}({\varphi }(g))$ . Next we check that $\varepsilon $ is a natural transformation. If ${\varphi }\colon S\to T$ is an idempotent bijective homomorphism of inverse semigroups, then the computation in the previous paragraph shows that ${\mathcal G}({\varphi })(\varepsilon _S(s)) = {\mathcal G}({\varphi })(D(s)) = D({\varphi }(s))=\varepsilon _T({\varphi }(s))$ . This completes the proof.

We next show that our functors preserve injective and surjective maps. There is presumably a categorical description of these morphisms in our categories, but we give a direct proof.

Proposition 3.5. The functors $\Gamma _c$ and ${\mathcal G}$ preserve injective and surjective morphisms.

Proof. Let ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ be an open iso-unital functor. Suppose first that ${\varphi }$ is injective. Then, for $U\in \Gamma _c({\mathscr G})$ , we have that ${\varphi }(U)\in E(\Gamma _c({\mathscr H}))$ implies ${\varphi }(U)\subseteq {\mathscr H}^{(0)}$ and, hence, $U\subseteq {\mathscr G}^{(0)}$ by Lemma 3.2. Thus $U\in E(\Gamma _c({\mathscr G}))$ and we conclude that $\Gamma _c({\varphi })$ is injective by another application of Lemma 3.2.

Next suppose that ${\varphi }$ is a surjective and let $V\in \Gamma _c({\mathscr H})$ . Note that V is compact Hausdorff, being homeomorphic to the compact Hausdorff space ${\mathop {\boldsymbol d}}(V)$ . We claim that, for each $h\in V$ , there is a compact open neighborhood $V_h\subseteq V$ of h such that there is a continuous section $s\colon V_h\to {\mathscr G}$ of ${\varphi }$ . Choose $g\in {\mathscr G}$ with ${\varphi }(g)=h$ . Then, since ${\varphi ^{-1}}(V)$ is open and the compact open bisections form a basis for the topology of ${\mathscr G}$ , as ${\mathscr G}$ is ample, we can find a compact open bisection $U_g$ with $g\in U_g$ and ${\varphi }(U_g)\subseteq V$ . Then $V_h={\varphi }(U_g)$ is a compact open set containing h (since ${\varphi }$ is open). Moreover, since ${\varphi }$ is iso-unital and $U_g$ is a bisection, ${\varphi }|_{U_g}$ is injective by Proposition 3.1. Then $s=({\varphi }|_{U_g})^{-1}$ is our desired section, as ${\varphi }$ is open.

Since V is compact, we can cover V by finitely many compact open subsets $V_1,\ldots , V_n$ such that there is a continuous section $s_i\colon V_i\to {\mathscr G}$ of ${\varphi }$ . Since V is compact Hausdorff, its compact open subsets form a Boolean algebra. The finitely many compact open subsets $V_1,\ldots , V_n$ of V generate a finite Boolean algebra with maximum element $V=\bigcup _{i=1}^nV_i$ . Let $W_1,\ldots , W_k$ be the atoms of this Boolean algebra. Then the $W_i$ are pairwise disjoint. Also since $W_i=V\cap W_i=(V_1\cap W_i)\cup \cdots \cup (V_n\cap W_i)$ , we see, using that $W_i$ is an atom, that $W_i\subseteq V_j$ for some j. Hence, there is a continuous section $t_i=s_j|_{W_i}\colon W_i\to {\mathscr G}$ of ${\varphi }$ . Since the maximum of a finite Boolean algebra is the join of its atoms, we deduce that $V=W_1\cup \cdots \cup W_k$ ; moreover this is a disjoint union. Hence, we can define a continuous section $s\colon V\to {\mathscr G}$ by putting $s|_{W_i}=t_i$ for $i=1,\ldots , k$ . Note that, since ${\varphi }$ is a local homeomorphism by Proposition 3.1, every continuous section of ${\varphi }$ defined on an open subset of ${\mathscr H}$ is an open mapping. Put $U=s(V)$ and note that U is compact open and ${\varphi }(U)=V$ . Moreover, ${\varphi }|_U$ is injective as ${\varphi }\circ s=1_V$ . We claim that $U\in \Gamma _c({\mathscr G})$ . If $g,g'\in U$ satisfy ${\mathop {\boldsymbol d}}(g)={\mathop {\boldsymbol d}}(g')$ , then ${\mathop {\boldsymbol d}}({\varphi }(g))={\mathop {\boldsymbol d}}({\varphi }(g'))$ and so ${\varphi }(g)={\varphi }(g')$ since V is a bisection. Therefore, $g=g'$ as ${\varphi }|_U$ is injective. Similarly, ${\mathop {\boldsymbol r}}|_U$ is injective and so $U\in \Gamma _c({\mathscr G})$ . Thus $\Gamma _c({\varphi })$ is surjective.

Suppose now that ${\varphi }\colon S\to T$ is an idempotent bijective homomorphism of Boolean inverse semigroups. First assume that ${\varphi }$ is injective and that $[{\varphi }(s),\chi \circ ({\varphi }|_{E(S)})^{-1}] {=}\,{\mathcal G}({\varphi })([s,\chi ])\in {\mathcal G}(T)^{(0)}$ . Then there is an idempotent $f\in E(T)$ such that $f\leq {\varphi }(s)$ and $\chi \circ ({\varphi }|_{E(S)})^{-1} (f)=1$ . Putting $e=({\varphi }|_{E(S)})^{-1}(f)$ , we have $\chi (e)=1$ and ${\varphi }(se)={\varphi }(s)f=f$ . Since ${\varphi ^{-1}}(E(T))=E(S)$ by Lemma 3.2, $se\in E(S)$ and $se\leq s$ . Moreover, $\chi (es^*se) =1$ and so $[s,\chi ]=[se,\chi ]\in {\mathcal G}^{(0)}$ . Therefore, ${\mathcal G}({\varphi })$ is injective by Lemma 3.2.

Next assume that ${\varphi }$ is surjective and let $[t,\chi ]\in {\mathcal G}(T)$ . Then $t={\varphi }(s)$ for some $s\in S$ and $\chi \circ {\varphi }|_{E(S)}(s^*s)=\chi (t^*t)=1$ . Hence, $[s,\chi \circ {\varphi }|_{E(S)}]\in {\mathcal G}(S)$ and ${\mathcal G}({\varphi })([s,\chi \circ {\varphi }|_{E(S)}]) = [{\varphi }(s),\chi ]=[t,\chi ]$ . Therefore, ${\mathcal G}({\varphi })$ is surjective.

Remark 3.6. The proof of Proposition 3.5 shows that if ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ is a surjective open iso-unital functor between ample groupoids, then a continuous local section of ${\varphi }$ can be defined on any compact open bisection of ${\mathscr H}$ .

Note that since $\Gamma _c$ and ${\mathcal G}$ are equivalences of categories, and isomorphisms in these categories are bijective, it follows that ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ is injective (respectively, surjective) if and only if $\Gamma _c({\varphi })\colon \Gamma _c({\mathscr G})\to \Gamma _c({\mathscr H})$ is injective (respectively, surjective), and ${\varphi }\colon S\to T$ is injective (respectively, surjective) if and only if ${\mathcal G}({\varphi })$ is injective (respectively, surjective).

The question of whether a surjective open iso-unital functor admits a continuous section that preserves unit spaces (but not necessarily a functor) is important when determining whether a twist comes from a groupoid $2$ -cocycle [5]. We show that, in the ample setting, this is equivalent to the question, considered in [15], of when a surjective idempotent bijective inverse semigroup homomorphism admits an order-preserving and idempotent-preserving section.

The following is a minor variation of [15, Lemma 4.2].

Lemma 3.7. Let ${\varphi }\colon S\to T$ be an idempotent bijective surjective homomorphism of inverse semigroups. Then the following are equivalent.

1. (1) There is an order-preserving map $j\colon T\to S$ with $j(E(T))\subseteq E(S)$ (that is, j is idempotent-preserving) and ${\varphi }\circ j=1_T$ .

2. (2) There is a map $j\colon T\to S$ that satisfies ${\varphi }\circ j=1_T$ and $j(te)=j(t)j(e)$ for all $t\in T$ and $e\in E(T)$ .

3. (3) There is a map $j\colon T\to S$ that satisfies ${\varphi }\circ j=1_T$ and $j(et)=j(e)j(t)$ for all $t\in T$ and $e\in E(T)$ .

Proof. We need only prove the equivalence of the first and second items as the equivalence of the first and third items follows by duality. Suppose first that j as in (2) exists. Then, if $e\in E(T)$ , we have $j(e) = j(ee)=j(e)j(e)$ and so j preserves idempotents. Also, if $t_1\leq t_2$ , we may write $t_1=t_2e$ for some $e\in E(T)$ . Therefore, $j(t_1)=j(t_2e) = j(t_2)j(e)\leq j(t_2)$ , as j preserves idempotents, and so j is order-preserving. Thus (2) implies (1).

Suppose that there is a map j satisfying the conditions specified in (1). Let $t\in T$ and $e\in E(T)$ . First we claim that $j(t)^*j(t)j(e)=j(te)^*j(te)$ . Indeed, applying ${\varphi }$ to both sides yields $t^*te$ . Consequently, $j(t)^*j(t)j(e)= j(te)^*j(te)$ as $j(e)\in E(S)$ and ${\varphi }$ is idempotent bijective. Since $te\leq t$ , the fact that j is order-preserving implies that $j(te)\leq j(t)$ and so

$$ \begin{align*} j(te)=j(t)j(te)^*j(te) = j(t)j(t)^*j(t)j(e)=j(t)j(e), \end{align*} $$

as required.

Proposition 3.8. Let ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ be a surjective open iso-unital functor between ample groupoids. There there is a continuous mapping $f\colon {\mathscr H}\to {\mathscr G}$ with ${\varphi }\circ f=1_{{\mathscr H}}$ and $f({\mathscr H}^{(0)})\subseteq {\mathscr G}^{(0)}$ if and only if there is an order-preserving and idempotent-preserving mapping $j\colon \Gamma _c({\mathscr H})\to \Gamma _c({\mathscr G})$ with $\Gamma _c({\varphi })\circ j=1_{\Gamma _c({\mathscr H})}$ .

Proof. Suppose first that f exists. Then, since ${\varphi }$ is a local homeomorphism by Proposition 3.1, the mapping f is open. Let $U\in \Gamma _c({\mathscr H})$ and put $j(U)=f(U)$ . Then $j(U)$ is compact open. It is also a bisection since, if $f(h),f(h')\in f(U)=j(U)$ satisfy ${\mathop {\boldsymbol d}}(f(h))={\mathop {\boldsymbol d}}(f(h'))$ and $h,h'\in U$ , then ${\mathop {\boldsymbol d}}(h)={\mathop {\boldsymbol d}}(h')$ because ${\varphi }$ is a functor and ${\varphi }\circ f=1_{{\mathscr H}}$ . It follows that $h=h'$ because U is a bisection. Similarly, ${\mathop {\boldsymbol r}}|_{j(U)}$ is injective and so $j(U)\in \Gamma _c({\mathscr G})$ . Finally, j is order-preserving and preserves idempotents since $f({\mathscr H}^{(0)})\subseteq {\mathscr G}^{(0)}$ .

For the converse, it suffices by Theorem 3.4 to show that if ${\varphi }\colon S\to T$ is a surjective idempotent bijective homomorphism of Boolean inverse semigroups that admits an order-preserving section $j\colon T\to S$ with $j(E(T))\subseteq E(S)$ , then ${\mathcal G}({\varphi })\colon {\mathcal G}(S)\to {\mathcal G}(T)$ admits a continuous section f that preserves units. Put $f([t,\chi ]) =[j(t),\chi \circ {\varphi }]$ . This is well defined since, if $[t,\chi ]=[t',\chi ]$ , we can find $u\leq t,t'$ with $\chi (u^*u)=1$ and so $j(u)\leq j(t),j(t')$ as j is order-preserving and $\chi ({\varphi }({\kern2pt}j(u)^*j(u))) = \chi (u^*u)=1$ . Thus $[j(t),\chi \circ {\varphi }]=[j(t'),\chi \circ {\varphi }]$ . Trivially, ${\mathcal G}({\varphi })(f([t,\chi ]))= {\mathcal G}({\varphi })([j(t),\chi \circ {\varphi }])= [{\varphi }({\kern2pt}j(t)),\chi ]=[t,\chi ].$ Also, if $[e,\chi ]\in {\mathcal G}(T)^{(0)}$ with $e\in E(T)$ , then $f([e,\chi ]) = [j(e),\chi \circ {\varphi }]\in {\mathcal G}(S)^{(0)}$ since $j(e)\in E(S)$ .

For continuity, we show that $f^{-1}(D(s)) = \bigcup _{j(t)\leq s}D(t)$ . Indeed, if $j(t)\leq s$ , then $f([t,\chi ]) = [j(t),\chi \circ {\varphi }]=[s,\chi \circ {\varphi }]\in D(s)$ . Conversely, if $[j(t),\chi \circ {\varphi }]=f([t,\chi ])\in D(s)$ , then there exists $u\leq j(t),s$ with $\chi ({\varphi }(u^*u))=1$ . Put $e=u^*u$ and note that $j(t)e=u=se$ . Then ${\varphi }(u)\leq {\varphi }({\kern2pt}j(t))=t$ and $\chi ({\varphi }(u)^*{\varphi }(u)) =1$ , and so $[t,\chi ]=[{\varphi }(u),\chi ]\in D({\varphi }(u))$ . Since ${\varphi }$ is idempotent bijective and j is idempotent-preserving, $j({\varphi }(e))=e$ . Thus $t{\varphi }(e)={\varphi }({\kern2pt}j(t)u^*u) = {\varphi }(u)$ , and so $j({\varphi }(u))=j(t{\varphi }(e)) = j(t)j({\varphi }(e))=j(t)e=se\leq s$ by Lemma 3.7. This completes the proof.

It is shown in [15, Proposition 4.6] that if ${\varphi }\colon S\to T$ is an idempotent bijective surjective homomorphism of Boolean inverse monoids whose idempotents form a complete Boolean algebra, then an order-preserving and idempotent-preserving section always exists. In the language of ample groupoids, these conditions mean that the unit spaces of ${\mathcal G}(S)$ and ${\mathcal G}(T)$ are Stonean (compact and extremally disconnected).

In [5] it was shown that twists over second countable Hausdorff ample groupoids admit a unit-preserving global section; in [4, Lemma 2.4] and the remark thereafter, it was observed that the same proof works for paracompact Hausdorff ample groupoids. The following proposition generalizes these results to open iso-unital functors.

Proposition 3.9. Let ${\varphi }\colon {\mathscr G}\to {\mathscr H}$ be a surjective open iso-unital functor between ample groupoids. If ${\mathscr H}$ is Hausdorff and ${\mathscr H}\setminus {\mathscr H}^{(0)}$ is paracompact, then there is a continuous section $s\colon {\mathscr H}\to {\mathscr G}$ with $s({\mathscr H}^{(0)})\subseteq {\mathscr G}^{(0)}$ .

Proof. Since ${\mathscr H}$ is Hausdorff, ${\mathscr H}^{(0)}$ is clopen in ${\mathscr H}$ . Therefore, ${\mathscr H}'={\mathscr H}\setminus {\mathscr H}^{(0)}$ is clopen. We show that every $\sigma $ -compact clopen subset X of ${\mathscr H}'$ admits a continuous section $s_X\colon X\to {\mathscr G}$ . Since X is a countable union of compact sets and each of these compact sets can be covered by finitely many compact open bisections of ${\mathscr H}$ contained in X, as ${\mathscr H}$ is ample and X is clopen, $X=\bigcup _{n=1}^\infty U_n$ , where the $U_n$ are compact open bisections of ${\mathscr H}$ . Put $V_1=U_1$ and $V_n = U_n\setminus \bigcup _{i=1}^{n-1}V_i$ . Then, since ${\mathscr H}$ is Hausdorff, each $V_n$ is a compact open bisection and, by construction, $\bigcup _{n=1}^\infty V_n=\bigcup _{n=1}^\infty U_n=X$ . Moreover, the $V_n$ are pairwise disjoint. By Remark 3.6, there is a continuous section $s_n\colon V_n\to {\mathscr G}$ of ${\varphi }$ and we can then define $s_X$ by $s_X|_{V_n}=s_n$ for $n\geq 1$ .

Next we use the well-known fact that a locally compact Hausdorff space is paracompact if and only if it can be partitioned into clopen $\sigma $ -compact subspaces to write ${\mathscr H}'=\coprod _{\alpha \in A}X_{\alpha }$ with each $X_{\alpha }$ clopen and $\sigma $ -compact. We now define $s\colon {\mathscr H}\to {\mathscr G}$ by $s|_{{\mathscr H}^{(0)}} = ({\varphi }|_{{\mathscr G}^{(0)}})^{-1}$ and $s|_{X_{\alpha }}=s_{X_{\alpha }}$ . This completes the proof.

3.2 Extensions of inverse semigroups

By an extension of inverse semigroups we mean a sequence

(3-1) $$ \begin{align} K\xrightarrow{\,\,\iota\,\,} T\xrightarrow{\,\,{\varphi}\,\,} S \end{align} $$

of idempotent bijective homomorphisms with $\iota (K)=\ker {\varphi }={\varphi ^{-1}}(E(S))$ and ${\varphi }(T)=S$ . We call the extension abelian if K is commutative.

Abelian extensions are classified by Lausch’s second cohomology group; see [20, Section 7]. We recall the setup. Let S be an inverse semigroup. An S-module consists of a commutative inverse semigroup K, an idempotent bijective homomorphism $p\colon K\to E(S)$ and a (total) left action of S on K (by endomorphisms) such that $p(sk) = sp(k)s^*$ and $p(k)k=k$ for all $k\in K$ and $s\in S$ . The category of S-modules is an abelian category with enough projectives and injectives. Lausch developed a corresponding cohomology theory based on the derived functors of the functor taking an S-module $p\colon K\to E(S)$ to the S-equivariant sections $q\colon E(S)\to K$ of p (with respect to the conjugation action on $E(S))$ [20]. We only need the second cohomology group, which classifies extensions. Note that Lausch uses right modules, so we have dualized his results here.

Given an extension (3-1) of S by a commutative inverse semigroup K, we can define an S-module structure on K by putting $p={\varphi }\iota \colon K\to E(S)$ , choosing a set-theoretic section $j\colon S\to T$ and setting

$$ \begin{align*} sk = \iota^{-1}({\kern2pt}j(s)\iota(k)j(s)^*) \end{align*} $$

for $s\in S$ and $k\in K$ . Note that $\iota (K)=\ker {\varphi }$ is a normal inverse subsemigroup of T, and so this makes sense. Lausch shows that the module structure is independent of the choice of j. Also, each element of $t\in T$ can be uniquely written as $t=\iota (k)j(s)$ with $s\in S$ and $k\in K$ , namely, $s={\varphi }(t)$ and $k =\iota ^{-1} (tj({\varphi }(t))^*)$ .

Given an S-module K with $p\colon K\to E(S)$ , a $2$ -cocycle is a mapping $c\colon S\times S\to K$ satisfying the following properties:

1. (1) $p(c(s,t)) = stt^*s^*$ ;

2. (2) $(sc(t,u))c(s,tu) = c(s,t)c(st,u)$ for $s,t,u\in S$ .

We say that c is normalized if $c(e,e)\in E(K)$ for all $e\in E(S)$ . The trivial $2$ -cocycle is defined by $c(s,t)=(p|_{E(K)})^{-1}(stt^*s^*)$ . The $2$ -cocycles form an abelian group under pointwise multiplication with the trivial cocycle as the identity and pointwise inversion: $c^{-1}(s,t) = c(s,t)^*$ . The group of $2$ -cocycles will be denoted $Z^2(S,K)$ . The set of all mappings $F\colon S\to K$ with $p(F(s))=ss^*$ is an abelian group $C^1(S,K)$ under pointwise operations, with identity $F(s) = (p|_{E(K)})^{-1} (ss^*)$ . There is a coboundary homomorphism $\delta \colon C^1(S,K)\to Z^2(S,K)$ given by $(\delta F)(s,t) = F(s)(sF(t))F(st)^*$ and the image $B^2(S,K)$ is the group of $2$ -coboundaries. We put $H^2(S,K)=Z^2(S,K)/B^2(S,K)$ and call it the second Lausch cohomology group of S with coefficients in K (in Lausch’s cohomology theory he calls this $H^2(S^I,K^0)$ where $S^I$ and $K^0$ are obtained by adjoining identities (see [20, Section 7]), but we avoid this extra notation for simplicity).

Proposition 3.10. Every $2$ -cocycle is cohomologous to a normalized one.

Proof. Let $c\colon S\times S\to K$ be a $2$ -cocycle. Define $F\colon S\to K$ by $F(s) = c(ss^*,ss^*)^*$ and put $c'=c\delta F$ . Then, for $e\in E(S)$ , we have $c'(e,e) = c(e,e)F(e)(eF(e))F(e^2)^* = c(e,e)c(e,e)^*c(e,e)^*c(e,e)\in E(K)$ . Thus $c'$ is normalized.

Given an abelian extension as per (3-1), we may choose a set-theoretic section $j\colon S\to T$ such that $j|_{E(S)} = ({\varphi }|_{E(T)})^{-1}$ . We can then define $c\colon S\times S\to K$ by $c(s,s') = \iota ^{-1}({\kern2pt}j(s)j(s')j(ss')^*)\in K$ ; equivalently, $j(s)j(s')=\iota (c(s,s'))j(ss')$ for all $s,s'\in S$ . One checks (compare [20, Section 7]) that c is a $2$ -cocycle. Moreover, it is normalized since $c(e,e) = \iota ^{-1} ({\kern2pt}j(e)j(e)j(ee)^\ast )=\iota ^{-1}({\kern2pt}j(e))\in E(K)$ , as $j(e)\in E(T)$ . Changing the section j results in a cohomologous $2$ -cocycle. Lausch does not require j to preserve idempotents, which results in a not necessarily normalized $2$ -cocycle, but we find normalized $2$ -cocycles more convenient to work with. So, for each extension of K by S, we obtain an S-module structure on K and a normalized $2$ -cocycle whose cohomology class is well defined.

Two extensions

$$ \begin{gather*} K\xrightarrow{\,\,\iota\,\,} T\xrightarrow{\,\,{\varphi}\,\,} S,\quad K\xrightarrow{\,\,\iota'\,\,} T'\xrightarrow{\,\,{\varphi}\,\,} S \end{gather*} $$

are equivalent if there is an isomorphism $\psi \colon T\to T'$ such that the diagram

commutes.

Lausch proves that two extensions of K by S are equivalent if and only if the module structures on K are the same and the corresponding cohomology classes of $2$ -cocycles are the same. Moreover, for each $2$ -cocycle, he shows that there is an extension (3-1) of K by S realizing the cohomology class of the $2$ -cocycle. If $c\colon S\times S\to K$ is a normalized $2$ -cocycle, we can put $T=\{(k,s)\in K\times S\mid p(k)=ss^*\}$ with multiplication given by $(k,s)(k',s') = (k (sk')c(s,s'),ss')$ . Here, $\iota (k) = (k,p(k))$ and ${\varphi }(k,s)=s$ . The inverse is given by $(k,s)^*=((s^*k^*)c(s,s^*)^*,s^*)$ . The class of the trivial cocycle corresponds to the split extension of K by S and T, which, in this case, is what is termed the full restricted semidirect product $K\bowtie S$ of K by S in [21, Ch. 5.3]. More details can be found in [20, Section 7].

We record here some properties of normalized $2$ -cocycles for later use.

Proposition 3.11. Let K be an S-module with idempotent bijective homomorphism $p\colon K\to E(S)$ and $c\colon S\times S\to K$ a normalized $2$ -cocycle.

1. (1) $c(s,s^*s)=c(ss^*,s)\in E(K)$ .

2. (2) $c(s,s^*) = sc(s^*,s)$ for all $s\in S$ .

3. (3) $c(e,ef)= c(ef,e)\in E(K)$ for all $e,f\in E(S)$ .

4. (4) $c(e,f)\in E(K)$ for all $e,f\in E(S)$ .

5. (5) $c(s,e)=c(s,s^*se)$ for all $e\in E(S)$ , $s\in S$ .

6. (6) $c(e,s) = c(ess^*,s)$ for all $e\in E(S)$ , $s\in S$ .

7. (7) $c(e,es), c(se,e)\in E(K)$ for all $e\in E(S)$ and $s\in S$ .

8. (8) $c(e,s) = c(s,s^*es)$ for all $e\in E(S)$ and $s\in S$ .

9. (9) $c(s,e) = c(ses^*,s)$ for all $e\in E(S)$ and $s\in S$ .

10. (10) $c(u,s)c(ut,s^*u^*us)^* = (uc(t,s^*s)^*)c(u,t)$ if $s\leq t\in S$ and $u\in S$ .

11. (11) $c(s,u)c(tu,u^*s^*su)^* = c(t,s^*s)^*c(t,u)$ if $s\leq t\in S$ and $u\in S$ .

Proof. For $e\in E(S)$ , let $k_e$ denote the unique idempotent of K with $p(k_e)=e$ . By the $2$ -cocycle condition, the definition of a module and since c is normalized, $c(s,s^*s)= (sc(s^*s,s^*s))c(s,s^*s)=c(s,s^*s)c(s,s^*s)$ . Also, the $2$ -cocycle condition and the definition of a module yield

$$ \begin{align*} c(ss^*,s)^2=(ss^*c(ss^*,s))c(ss^*,s) =c(ss^*,ss^*)c(ss^*,s)=c(ss^*,s), \end{align*} $$

as c is normalized. Thus $c(s,s^*s),c(ss^*,s)$ are idempotents with image $ss^*$ under p and hence are equal as p is idempotent bijective. This yields (1).

For (2), we have $(sc(s^*,s))c(s,s^*s)=c(s,s^*)c(ss^*,s)$ . By the first item, we conclude that $c(s,s^*)=sc(s^*,s)$ .

For (3), we compute $c(e,ef)^2 = (ec(e,ef))c(e,ef) = c(e,e)c(e,ef)=c(e,ef)$ since c is normalized and

$$ \begin{align*} c(e,ef)=efc(e,ef)=ec(e,ef)=c(e,e)c(e,ef). \end{align*} $$

Similarly, we have

$$ \begin{align*} c(ef,e)=(efc(e,e))c(ef,e)=c(ef,e)c(ef,e) \end{align*} $$

as $efc(e,e)=k_{ef}$ because c is normalized. Since $p(c(e,ef))=ef=p(c(ef,e))$ and p is idempotent bijective, we deduce that $c(e,ef)=c(ef,e)$ .

To prove (4), observe that $(ec(f,ef))c(e,ef) =c(e,f)c(ef,ef) =c(e,f)$ since c is normalized, and so $c(e,f)\in E(K)$ by (3).

For (5), we have $(sc(s^*s,e))c(s,s^*se) = c(s,s^*s)c(s,e)$ . By (4), note that $sc(s^*s,e) = k_{ses^*}$ and, by (1), $c(s,s^*s) = k_{ss^*}\geq k_{ses^*}$ , and so the left-hand side is $c(s,s^*se)$ and the right-hand side is $c(s,e)$ .

For (6), we have $(ec(ss^*,s))c(e,s)=c(e,ss^*)c(ess^*,s)$ . As $c(ss^*,s)=k_{ss^*}$ , $c(e,ss^*)=k_{ess^*}$ by (1) and (4), and we deduce that $c(e,s)=c(ess^*,s)$ .

For (7), we have $c(e,es) = c(ess^*,es)\in E(K)$ and $c(se,e)=c(se,es^*s)\in E(K)$ by (6) and (1).

For (8), we note by the $2$ -cocycle condition that

$$ \begin{align*} (ec(s,s^*es))c(e,es)=c(e,s)c(es,s^*es). \end{align*} $$

But by (1), $c(es,s^*es)=k_{ess^*}$ and by (7), $c(e,es)=k_{ess^*}$ . Also, since $p(c(s,s^*es)) = ess^*\leq e$ , we have

$$ \begin{align*} ec(s,s^*es)=ess^*c(s,s^*es)=c(s,s^*es). \end{align*} $$

We conclude that $c(s,s^*es)=c(e,s)$ as required.

For (9), we note that $(ses^*c(s,e))c(ses^*,se) = c(ses^*,s)c(se,e)$ by the $2$ -cocycle condition. But $c(ses^*,se)=k_{ses^*}=c(se,e)$ by (1) and (7), and so we deduce, since $p(c(s,e)) = ses^*$ , that $c(s,e)=c(ses^*,s)$ .

To prove (10), we compute

$$ \begin{align*} (uc(t,s^*s))c(u,s) &= (uc(t,s^*s))c(u,ts^*s) = c(u,t)c(ut,s^*s)\\[-2pt] &= c(u,t)c(ut, t^*u^*uts^*s) \end{align*} $$

where the last equality uses (5). But $t^*u^*uts^*s=s^*st^*u^*uts^*s=s^*u^*us$ . Thus we have $(uc(t,s^*s))c(u,s)=c(u,t)c(ut,s^*u^*us)$ . Using $p(uc(t,s^*s)) = uss^*u^*=p(c(ut,s^*u^*us))$ , we deduce that

$$ \begin{align*} c(u,s)c(ut,s^*u^*us)^* = (uc(t,s^*s)^*)c(u,t), \end{align*} $$

as required.

We now turn to (11). From the $2$ -cocycle condition we obtain

$$ \begin{align*} (tc(u,u^*s^*su))c(t,s^*su) &= c(t,u)c(tu,u^*s^*su), \\[-2pt] (tc(s^*s,u))c(t,s^*su) &= c(t,s^*s)c(ts^*s,u) \\[-2pt] &= c(t,s^*s)c(s,u). \end{align*} $$

Note that $c(s^*s,u) = c(u,u^*s^*su)$ by (8). Therefore, $c(t,s^*s)c(s,u)= c(t,u)c(tu,u^*s^*su)$ . Then, using

$$ \begin{align*} p(c(t,s^*s))&=ss^*\geq suu^*s^*=p(c(tu,u^*s^*su))\\[-2pt] &=p(c(s,u)) = p(c(t,s^*s)^*c(t,u)), \end{align*} $$

we deduce that $c(s,u)c(tu,u^*s^*su)^* = c(t,s^*s)^*c(t,u)$ , as required.

3.3 Twists and extensions of groupoids

By an extension of ample groupoids we mean an exact sequence

(3-2) $$ \begin{align} {\mathscr K}\xrightarrow{\,\,\iota\,\,} {\mathscr H}\xrightarrow{\,\,{\varphi}\,\,} {\mathscr G} \end{align} $$

of open iso-unital functors with $\iota $ injective, ${\varphi }$ surjective and ${\varphi }^{-1}({\mathscr G}^{(0)}) = \iota ({\mathscr K})$ . A second extension

$$ \begin{align*} {\mathscr K}\xrightarrow{\,\,\iota'\,\,} {\mathscr H}'\xrightarrow{\,\,{\varphi}'\,\,} {\mathscr G} \end{align*} $$

is equivalent to (3-2) if there is an isomorphism $\psi \colon {\mathscr H}\to {\mathscr H}'$ making the diagram

commute.

We now show that our functors $\Gamma _c$ and ${\mathcal G}$ are exact in the sense that they preserve extensions.

Theorem 3.12. The functors $\Gamma _c$ and ${\mathcal G}$ preserve extensions. More precisely, we have the following statements.

1. (1) If ${\mathscr K}\xrightarrow {\,\,\iota \,\,} {\mathscr H}\xrightarrow {\,\,{\varphi }\,\,} {\mathscr G}$ is an extension of ample groupoids, then

   $$ \begin{align*} \Gamma_c({\mathscr K})\xrightarrow{\,\,\Gamma_c(\iota)\,\,} \Gamma_c({\mathscr H})\xrightarrow{\,\,\Gamma_c({\varphi})\,\,} \Gamma_c({\mathscr G}) \end{align*} $$
   is an extension of Boolean inverse semigroups.

2. (2) If $K\xrightarrow {\,\,\iota \,\,} T\xrightarrow {\,\,{\varphi }\,\,} S$ is an extension of Boolean inverse semigroups, then

   $$ \begin{align*} {\mathcal G}(K)\xrightarrow{\,\,{\mathcal G}(\iota)\,\,} {\mathcal G}(T)\xrightarrow{\,\,{\mathcal G}({\varphi})\,\,} {\mathcal G}(S) \end{align*} $$
   is an extension of ample groupoids.

Proof. By Theorem 3.4 and Proposition 3.5, all that remains to check is that $\Gamma _c(\iota )(\Gamma _c({\mathscr K}))=\Gamma _c({\varphi })^{-1}(E(\Gamma _c({\mathscr G})))$ and ${\mathcal G}(\iota )({\mathcal G}(K)) = {\mathcal G}({\varphi })^{-1} ({\mathcal G}(S)^{(0)})$ . For the first item, note that, since ${\mathscr G}^{(0)}$ is open, ${\varphi }^{-1}({\mathscr G}^{(0)})$ is an open subgroupoid of ${\mathscr H}$ and, hence, ample. Moreover, $\Gamma _c({\varphi })^{-1}(E(\Gamma _c({\mathscr G})))=\Gamma _c({\varphi ^{-1}}({\mathscr G}^{(0)}))$ . Since $\iota \colon {\mathscr K}\to {\varphi }^{-1}({\mathscr G}^{(0)})$ is surjective and open iso-unital, we deduce from Proposition 3.5 that

$$ \begin{align*} \Gamma(\iota)(\Gamma_c({\mathscr K})) = \Gamma_c({\varphi}^{-1}({\mathscr G}^{(0)}))=\Gamma_c({\varphi})^{-1}(E(\Gamma_c({\mathscr G}))), \end{align*} $$

as required.

For the second item, let $[k,\chi ]\in {\mathcal G}(K)$ . Then

$$ \begin{align*} {\mathcal G}({\varphi}){\mathcal G}(\iota)([k,\chi]) = [{\varphi}\iota(k),\chi\circ ({\varphi}\iota|_{E(K)})^{-1}]\in {\mathcal G}(S)^{(0)} \end{align*} $$

since ${\varphi }\iota (k)\in E(S)$ . Conversely, if $[{\varphi }(t),\chi \circ ({\varphi }|_{E(T)})^{-1}]={\mathcal G}({\varphi })([t,\chi ])\in {\mathcal G}(S)^{(0)}$ , then there is an idempotent $f\in E(S)$ with $f\leq {\varphi }(t)$ and

$$ \begin{align*} \chi\circ ({\varphi}|_{E(S)})^{-1}(f)=1. \end{align*} $$

Putting $e=({\varphi }|_{E(S)})^{-1} (f)$ , we have that $\chi (e)=1$ and ${\varphi }(te)={\varphi }(t)f=f$ . Since ${\chi (et^*te)=1}$ , we deduce that $[t,\chi ]=[te,\chi ]$ . As $te\in {\varphi ^{-1}}(E(S))$ , we have that $te=\iota (k)$ for some $k\in K$ . Then $[k,\chi \circ \iota |_{E(K)}]\in {\mathcal G}(K)$ (as $\chi \iota (k^*k) = \chi (et^*te)=1$ ) and ${\mathcal G}(K)(\iota )([k,\chi \circ \iota |_{E(K)}]) = [te,\chi ]=[t,\chi ]$ . Thus ${\mathcal G}(\iota )({\mathcal G}(K))= {\mathcal G}({\varphi })^{-1}({\mathcal G}(S)^{(0)})$ , as required. This completes the proof.

It follows from Theorem 3.12 that classifying extensions of ample groupoids is equivalent to classifying extensions of Boolean inverse semigroups. One has to take some care when applying the Lausch cohomology in order to make sure that the extension coming from a $2$ -cocycle is Boolean.

We briefly digress to examine how the Hausdorff property behaves under extensions of ample groupoids. It follows from the results of [38] (see also [22]) that an ample groupoid ${\mathscr G}$ is Hausdorff if and only if each $U\in \Gamma _c({\mathscr G})$ has a maximum idempotent below it in the natural partial order. When ${\mathscr G}$ is Hausdorff, $U\cap {\mathscr G}^{(0)}$ is the maximum idempotent below U.

Proposition 3.13. Consider an extension of ample groupoids as in (3-2). Then ${\mathscr G}$ is Hausdorff if and only if each element of $\Gamma _c({\mathscr H})$ has a unique maximal element of $\Gamma _c(\iota )(\Gamma _c({\mathscr K}))$ below it. Moreover, if ${\mathscr G}$ and ${\mathscr K}$ are both Hausdorff, then so is ${\mathscr H}$ .

Proof. Without loss of generality, we may assume that $\iota $ is an inclusion. Suppose that ${\mathscr G}$ is Hausdorff and $U\in \Gamma _c({\mathscr H})$ . Then ${\varphi }(U)\in \Gamma _c({\mathscr G})$ has a unique maximum idempotent $V\subseteq {\varphi }(U)$ . Since $V\subseteq {\mathscr G}^{(0)}$ , we have ${\varphi ^{-1}}(V)\subseteq {\mathscr K}$ . Also, since V is clopen (as ${\mathscr G}$ is Hausdorff), ${\varphi ^{-1}}(V)\cap U\subseteq {\mathscr K}$ is clopen and, hence, compact (since U is compact). Thus $W={\varphi ^{-1}}(V)\cap U\in \Gamma _c({\mathscr K})$ and $W\subseteq U$ . Moreover, if $W'\in \Gamma _c({\mathscr K})$ with $W'\subseteq U$ , then ${\varphi }(W')\subseteq {\varphi }(U)\cap {\mathscr G}^{(0)}=V$ and so $W'\subseteq W$ . Thus W is the maximum element of $\Gamma _c({\mathscr K})$ below U.

For the converse, by Proposition 3.5, every element of $\Gamma _c({\mathscr G})$ is of the form ${\varphi }(U)$ for some $U\in \Gamma _c({\mathscr H})$ . Let W be the maximum element of $\Gamma _c({\mathscr K})$ below U. Then ${\varphi }(W)\subseteq {\mathscr G}^{(0)}$ is an idempotent and ${\varphi }(W)\subseteq {\varphi }(U)$ . Suppose that $V\subseteq {\mathscr G}^{(0)}$ is compact open with $V\subseteq {\varphi }(U)$ . Let $x\in V$ and let $h\in U$ satisfy ${\varphi }(h)=x$ . Since $U\cap {\varphi ^{-1}}(V)$ is an open subset of ${\mathscr K}$ containing h, we can find a compact open bisection $W'\in \Gamma _c({\mathscr K})$ with $h\in W'\subseteq U\cap {\varphi ^{-1}}(V)\subseteq {\mathscr K}$ . Then $W'\subseteq W$ by choice of W and so ${\varphi }(W')\subseteq {\varphi }(W)$ . As $x={\varphi }(h)\in {\varphi }(W')$ , we deduce that $x\in {\varphi }(W)$ and, hence, $V\subseteq {\varphi }(W)$ . This shows that ${\varphi }(W)$ is the maximum idempotent below ${\varphi }(U)$ and, hence, ${\mathscr G}$ is Hausdorff.

Suppose now that ${\mathscr G}$ and ${\mathscr K}$ are Hausdorff. We continue to assume that $\iota $ is an inclusion. If $U\in \Gamma _c({\mathscr H})$ , then there is a unique maximum element V of $\Gamma _c({\mathscr K})$ with $V\subseteq U$ by the above paragraph. Since ${\mathscr K}$ is Hausdorff, there is a unique maximum idempotent $W\subseteq V$ in $\Gamma _c({\mathscr K})$ . Note that $E(\Gamma _c({\mathscr K})) = E(\Gamma _c({\mathscr H}))$ as ${\mathscr K}^{(0)}={\mathscr H}^{(0)}$ . Thus any idempotent $U'\subseteq U$ belongs to $\Gamma _c({\mathscr K})$ , and, hence, is below V, and therefore below W. Thus W is the maximum idempotent below U. It follows that ${\mathscr H}$ is Hausdorff.

Let ${\mathscr G}$ be an ample groupoid and A a discrete abelian group with identity $1$ , whose binary operation is written multiplicatively (we can think of A of as a one-object ample groupoid). In practice, $A=R^\times $ , where R is a commutative ring with unity. A (discrete) A-twist over ${\mathscr G}$ is an exact sequence of ample groupoids

(3-3) $$ \begin{align} A\times {\mathscr G}^{(0)}\xrightarrow{\,\,\iota\,\,} \Sigma\xrightarrow{\,\,{\varphi}\,\,} {\mathscr G} \end{align} $$

(with ${\varphi }\iota (a,x)=x$ ) which is central in the sense that

(3-4) $$ \begin{align} \iota(a,{\mathop{\boldsymbol r}}({\varphi}(s)))s=s\iota(a,{\mathop{\boldsymbol d}}({\varphi}(s))) \end{align} $$

for all $a\in A$ and $s\in \Sigma $ . In this case, we often write $as$ for $\iota (a,{\mathop {\boldsymbol r}}({\varphi }(s)))s=s\iota (a,{\mathop {\boldsymbol d}}({\varphi }(s)))$ and we note that $(as)(a's') = (aa')ss'$ for $a,a'\in A$ and $s,s'\in \Sigma $ with ${\mathop {\boldsymbol d}}(s)={\mathop {\boldsymbol r}}(s')$ . Also note that $(a,s)\mapsto as$ is a free action of A on $\Sigma $ by homeomorphisms (but not by groupoid automorphisms). Note that since $A\times {\mathscr G}^{(0)}$ is Hausdorff, if ${\mathscr G}$ is Hausdorff, then so is $\Sigma $ by Proposition 3.13. This was already observed in [4, Corollary 2.3] using an argument specific to twists.

Note that a different definition of twists is given in [5], where $\Sigma $ is not required to be ample and $\iota $ is not required to be open, but where ${\varphi }$ is required to be a locally trivial fiber bundle with fiber A. However, it is shown in [4, Proposition 2.2] that our definition is equivalent to the one in [5]. The first step in reformulating A-twists in terms of inverse semigroups is to identify $\Gamma _c(A\times {\mathscr G}^{(0)})$ in semigroup-theoretic terms. Note that $A\cup \{0\}$ is an inverse semigroup, where $0a=0=a0=0^2$ for all $a\in A$ . In addition, $a^*=a^{-1}$ if $a\in A$ and $0^*=0$ .

Proposition 3.14. Let $\widetilde {A} =C_c({\mathscr G}^{(0)},A\cup \{0\})$ be the set of locally constant mappings $f\colon {\mathscr G}^{(0)}\to A\cup \{0\}$ with compact support (meaning ${\mathop {\mathrm {supp}}}(f)=f^{-1}(A)$ is compact). Then $\widetilde {A}$ is a commutative inverse semigroup under pointwise multiplication and there is an isomorphism $\gamma \colon \widetilde {A}\to \Gamma _c(A\times {\mathscr G}^{(0)})$ given by $f\mapsto U_f=\{(f(x),x)\mid x\in {\mathop {\mathrm {supp}}}(f)\}$ .

Proof. Clearly $\widetilde {A}$ is a commutative inverse semigroup with $f^*(x) = f(x)^\ast $ giving the inversion. Note that

$$ \begin{align*} {\mathop{\mathrm{supp}}}(fg) = {\mathop{\mathrm{supp}}}(f)\cap {\mathop{\mathrm{supp}}}(g)={\mathop{\mathrm{supp}}}(f){\mathop{\mathrm{supp}}}(g) \end{align*} $$

since A is a group. The map $\gamma $ is a homomorphism because

$$ \begin{align*} U_fU_g &= \{(f(x),x)(g(x),x)\mid x\in {\mathop{\mathrm{supp}}}(f)\cap {\mathop{\mathrm{supp}}}(g)\} \\ &= \{(f(x)g(x),x)\mid x\in {\mathop{\mathrm{supp}}}(fg)\}=U_{fg}. \end{align*} $$

In addition, $\gamma $ is injective since ${\mathop {\mathrm {supp}}}(f)={\mathop {\boldsymbol d}}(U_f)$ and $f = \pi \circ ({\mathop {\boldsymbol d}}|_{U_f})^{-1}$ on its support, where $\pi $ is the projection to A. Conversely, given $U\in \Gamma _c(A\times {\mathscr G}^{(0)})$ , we can define f by

$$ \begin{align*} f(x) = \begin{cases}\pi\circ ({\mathop{\boldsymbol d}}|_U)^{-1} (x) & \text{if } x\in {\mathop{\boldsymbol d}}(U),\\ 0 & \text{otherwise},\end{cases} \end{align*} $$

and $U=U_f$ . Thus $\gamma $ is an isomorphism.

We can make $\widetilde {A}$ into a $\Gamma _c({\mathscr G})$ -module as follows. Define $p\colon \widetilde {A}\to E(\Gamma _c({\mathscr G}))$ by $p(f)= {\mathop {\mathrm {supp}}}(f)$ . The action is given by

$$ \begin{align*} (Uf)(x) = \begin{cases} f({\mathop{\boldsymbol d}}(({\mathop{\boldsymbol r}}|_U)^{-1}(x))) & \text{if } x\in {\mathop{\boldsymbol r}}(U),\\ 0 & \text{otherwise},\end{cases} \end{align*} $$

for $U\in \Gamma _c({\mathscr G})$ and $x\in {\mathscr G}^{(0)}$ .

Proposition 3.15. The commutative inverse semigroup $\widetilde {A}$ is a $\Gamma _c({\mathscr G})$ -module with respect to p and the action $(U,f)\mapsto Uf$ .

Proof. The idempotents of $\widetilde {A}$ are the characteristic functions $1_U$ with $U\subseteq {\mathscr G}^{(0)}$ compact open and $p(1_U)=U$ . Since ${\mathop {\mathrm {supp}}}(fg)={\mathop {\mathrm {supp}}}(f)\cap {\mathop {\mathrm {supp}}}(g)={\mathop {\mathrm {supp}}}(f){\mathop {\mathrm {supp}}}(g)$ , we have that p is an idempotent bijective semigroup homomorphism. If $W\subseteq {\mathscr G}^{(0)}$ is compact open and $U\in \Gamma _c({\mathscr G})$ , then $UWU^*$ consists of those $x\in {\mathscr G}^{(0)}$ such that there is an arrow $g\colon y\to x$ in U with $y\in W$ . The support of $Uf$ consists of those $x\in {\mathscr G}^{(0)}$ such that there is an arrow $g\colon y\to x$ with $f(y)\neq 0$ . Hence, ${\mathop {\mathrm {supp}}}(Uf) \!= U{\mathop {\mathrm {supp}}}(f)U^*$ . Also notice that if $U\subseteq {\mathscr G}^{(0)}$ is compact open, then $Uf \!=\! 1_Uf$ . It follows that $p(f)f= 1_{{\mathop {\mathrm {supp}}}(f)}f=f$ . It remains to check that $(U,f)\mapsto Uf$ is a semigroup action by endomorphisms.

Clearly, $U(fg)=(Uf)(Ug)$ from the definition. If $U,V\in \Gamma _c({\mathscr G})$ , then $x\in {\mathop {\mathrm {supp}}}(U(Vf))$ if and only if there is $g\colon y\to x$ with $g\in U$ and $h\in V$ with $h\colon z\to y$ for some z such that $f(z) \neq 0$ , in which case $(U(Vf))(x)\!=\!f(z)$ . But $(UV)f(x)$ is nonzero if and only if there is $k\colon z\to x$ in $UV$ for some z such that $f(z)\neq 0$ , in which case $(UV)f(x) \!=\! f(z)$ . But then $k=gh$ for unique $g\in U$ and $h\in V$ with, say, $g\colon y\to x$ and $h\colon z\to y$ . We deduce that $(UV)f(x) = U(Vf(x))$ . This concludes the proof that $\widetilde {A}$ is a $\Gamma _c({\mathscr G})$ -module.

Given an extension (3-3), the question of whether it is central (that is, satisfies (3-4)) can be determined from the corresponding $\Gamma _c({\mathscr G})$ -module structure on $\Gamma _c(A\times {\mathscr G}^{(0)})$ in the inverse semigroup extension

(3-5) $$ \begin{align} \Gamma_c(A\times{\mathscr G}^{(0)})\xrightarrow{\Gamma_c(\iota)} \Gamma_c(\Sigma)\xrightarrow{\Gamma_c({\varphi})} \Gamma_c({\mathscr G}). \end{align} $$

Proposition 3.16. An extension (3-3) is central if and only if the semigroup isomorphism $\gamma \colon \widetilde {A}\to \Gamma _c(A\times {\mathscr G}^{(0)})$ of Proposition 3.14 is an isomorphism of $\Gamma _c({\mathscr G})$ -modules with respect to the module structure on $\Gamma _c(A\times {\mathscr G}^{(0)})$ coming from the extension (3-5).

Proof. Note that ${\varphi }\iota (\gamma (f))={\varphi }(\iota (U_f)) = {\mathop {\mathrm {supp}}}(f)$ by construction. So we just need to check that $\Gamma _c({\mathscr G})$ -equivariance is equivalent to centrality. Fix a section $j\colon \Gamma _c({\mathscr G})\to \Gamma _c(\Sigma )$ of $\Gamma _c({\varphi })$ with $j|_{E(\Gamma _c({\mathscr G}))}= (\Gamma _c({\varphi })|_{E(\Gamma _c(\Sigma ))})^{-1}$ . Then the action of $V\in \Gamma _c({\mathscr G})$ on $\gamma (f)=U_f$ is given by

$$ \begin{align*} VU_f = \iota^{-1}({\kern2pt}j(V)\iota(U_f)j(V)^*). \end{align*} $$

This set consists of all $(a,x)$ such that there exist $y\in {\mathop {\mathrm {supp}}}(f)$ and $s\in j(V)$ with ${\varphi }(s)\colon y\to x$ and $(a,x)=\iota ^{-1}(s\iota (f(y),y)s^{-1})$ . On the other hand, $(Vf)(x)\neq 0$ if and only if there exist $g\in V$ and $y\in {\mathop {\mathrm {supp}}}(f)$ , with $g\colon y\to x$ ; and then $(Vf)(x)=f(y)$ . Since ${\varphi }|_{j(V)}\colon j(V)\to V$ is a homeomorphism by Proposition 3.1, this is equivalent to there being $s\in j(V)$ and $y\in {\mathop {\mathrm {supp}}}(f)$ with ${\varphi }(s)\colon y\to x$ and then $(Vf)(x)=f(y)$ . Thus $VU_f=U_{Vf}=\gamma (Vf)$ if and only if $\iota (f({\mathop {\boldsymbol d}}({\varphi }(s))),{\mathop {\boldsymbol r}}({\varphi }(s)))=s\iota (f({\mathop {\boldsymbol d}}({\varphi }(s))),{\mathop {\boldsymbol d}}({\varphi }(s)))s^{-1}$ for all $s\in j(V)$ with ${\mathop {\boldsymbol d}}({\varphi }(s))\in {\mathop {\mathrm {supp}}}(f)$ . This equivalent to

(3-6) $$ \begin{align} \iota(f({\mathop{\boldsymbol d}}({\varphi}(s))),{\mathop{\boldsymbol r}}({\varphi}(s)))s= s\iota(f({\mathop{\boldsymbol d}}({\varphi}(s))),{\mathop{\boldsymbol d}}({\varphi}(s))) \end{align} $$

for all $s\in j(V)$ with ${\mathop {\boldsymbol d}}({\varphi }(s))\in {\mathop {\mathrm {supp}}}(f)$ , which is satisfied if the extension is central. Hence, if the extension is central, $\gamma $ is an isomorphism of modules.

Conversely, if $\gamma $ is a module isomorphism and $s\in \Sigma $ , $a\in A$ , then we can choose a bisection V containing ${\varphi }(s)$ and we may assume that $j(V)$ was chosen to contain s. We can define $f = a1_{{\mathop {\boldsymbol d}}(V)}$ . Then, by (3-6), the equivariance of $\gamma $ implies that $\iota (a,{\mathop {\boldsymbol r}}({\varphi }(s)))s= \iota (f({\mathop {\boldsymbol d}}({\varphi }(s))),{\mathop {\boldsymbol r}}({\varphi }(s)))s =s\iota (f({\mathop {\boldsymbol d}} ({\varphi }(s))),{\mathop {\boldsymbol d}}({\varphi }(s))) = s\iota (a,{\mathop {\boldsymbol d}}({\varphi }(s)))$ . We conclude that the extension is central.

We may now deduce from Theorem 3.4, Theorem 3.12 and Proposition 3.16 the following corollary.

Corollary 3.17. There is a bijection between equivalence classes of A-twists over $\Gamma $ and extensions $\widetilde {A}\to T\to \Gamma _c({\mathscr G})$ with T a Boolean inverse semigroup and with the $\Gamma _c({\mathscr G})$ -module structure on $\widetilde {A}$ as per Proposition 3.15.

To complete our classification of A-twists by cohomology classes, we need every extension of $\widetilde {A}$ by $\Gamma _c({\mathscr G})$ to be a Boolean inverse semigroup. In fact, it turns out that every extension of a commutative Boolean inverse semigroup by a Boolean inverse semigroup is Boolean.

Proposition 3.18. Let S be a Boolean inverse semigroup and K a commutative Boolean inverse semigroup. If $K\xrightarrow {\,\,\iota \,\,} T\xrightarrow {\,\,{\varphi }\,\,} S$ is any extension of S by K, then T is a Boolean inverse semigroup.

Proof. Choose a section $j\colon S\to T$ with $j|_{E(S)} = ({\varphi }|_{E(T)})^{-1}$ . We can then turn K into an S-module (with $p={\varphi }\iota \colon K\to E(S)$ ); denote by $c\colon S\times S\to K$ the corresponding normalized $2$ -cocycle. Since $E(T)\cong E(S)$ is a Boolean algebra, we just need to show that if $t,u\in T$ are orthogonal, then they have a join. In fact, it is enough to show they have a common upper bound y for then, by [21, Proposition 1.4.18], one has $y(t^*t\vee u^*u) = yt^*t\vee yu^*u = t\vee u$ . Put $\widetilde {t}={\varphi }(t)$ and $\widetilde {u}={\varphi }(u)$ . Then the fact that $t,u$ are orthogonal is equivalent to $t^*t,u^*u$ and $tt^*,uu^*$ being orthogonal. This in turn is equivalent to $\widetilde {t},\widetilde {u}$ being orthogonal since ${\varphi }$ is idempotent bijective. Thus $x=\widetilde {t}\vee \widetilde {u}$ exists. We can write $t=\iota (k_t)j(\widetilde {t})$ and $u=\iota (k_u)j({\kern1pt}\widetilde {u}{\kern1pt})$ for unique $k_t,k_u\in K$ with $p(k_t) = \widetilde {t}{\kern4pt}\widetilde {t}^*$ and $p(k_u)=\widetilde {u}{\kern2pt}\widetilde {u}^*$ . Then $p(k_tc(x,\widetilde {t}^*{\kern2pt}\widetilde {t}{\kern1pt})^*) = \widetilde {t}{\kern4pt}\widetilde {t} ^*$ and $p(k_uc(x,\widetilde {u}^*{\kern2pt}\widetilde {u})^*)=\widetilde {u}{\kern2pt}\widetilde {u}^*$ and, hence, since p is idempotent bijective, $k=k_tc(x,\widetilde {t}^*{\kern2pt}\widetilde {t}{\kern1pt})^* \vee k_uc(x,\widetilde {u}^*{\kern2pt}\widetilde {u})^*$ is defined and $p(k)=\widetilde {t}{\kern4pt}\widetilde {t}^*\vee \widetilde {u}{\kern2pt}\widetilde {u}^*= xx^*$ (since p is additive, being idempotent bijective). Let $y=\iota (k)j(x)$ . We claim that $t,u\leq y$ . The argument is symmetric, so we just give it for t.

Note that $\iota (K)$ centralizes $E(T)$ as $\iota (k)e=\iota (kf)=\iota (fk)=e\iota (k)$ , where $f\in E(K)$ is unique with $\iota (f)=e$ (using that $\iota $ is idempotent bijective). We then have

$$ \begin{align*} yt^*t = \iota(k)j(x)j({\kern1pt}\widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt})=\iota(k)\iota(c(x,\widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt}))j(x\widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt}) = \iota(kc(x,\widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt}))j({\kern1pt}\widetilde{t}{\kern1pt}). \end{align*} $$

Therefore, it suffices to show that $kc(x,\widetilde {t}^*{\kern2pt}\widetilde {t}{\kern1pt}) = k_t$ . Note that $p(c(x,\widetilde {t}^*{\kern2pt}\widetilde {t}{\kern1pt}))= \widetilde {t}^*{\kern2pt}\widetilde {t}$ and so $k_uc(x,\widetilde {u}^*{\kern2pt}\widetilde {u}{\kern1pt})^*c(x,\widetilde {t}^*{\kern2pt}\widetilde {t}{\kern1pt}) =0$ . Therefore,

$$ \begin{align*} kc(x,\widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt}) = k_tc(x, \widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt})^*c(x, \widetilde{t}^*{\kern2pt}\widetilde{t}{\kern1pt}) =k_t, \end{align*} $$

as required, since multiplication distributes over joins in a Boolean inverse semigroup. This completes the proof.

Putting together Theorem 3.4, Theorem 3.12, Corollary 3.17, Proposition 3.18, and [20, Section 7], we obtain the following classification of A-twists.

Theorem 3.19. Let ${\mathscr G}$ be an ample groupoid and A a discrete abelian group. Then there is a bijection between equivalence classes of discrete A-twists and $H^2(\Gamma _c({\mathscr G}), C_c({\mathscr G}^{(0)}, A\cup \{0\}))$ .

Note that the class of the trivial $2$ -cocycle corresponds to the trivial extension $A\times {\mathscr G}^{(0)}\to A\times {\mathscr G}\to {\mathscr G}$ . Indeed, $\Gamma _c(A\times {\mathscr G})$ can be identified with the full restricted semidirect product $\widetilde {A}\bowtie \Gamma _c({\mathscr G})$ via the mapping that sends $(f,U)\in \widetilde {A}\bowtie \Gamma _c({\mathscr G})$ with ${\mathop {\mathrm {supp}}}(f)={\mathop {\boldsymbol r}}(U)$ to the compact open bisection $\{(f({\mathop {\boldsymbol r}}(g)),g)\mid g\in U\}$ of $A\times {\mathscr G}$ , as is easily checked. The abelian group structure on $H^2(\Gamma _c({\mathscr G}),C_c({\mathscr G}^{(0)}, A\cup \{0\}))$ corresponds to the Baer sum operation on A-twists over ${\mathscr G}$ . We state the result, but omit many of the routine verifications as they are not needed in the sequel.

If

are A-twists, then since ${\varphi }$ and ${\varphi }'$ are local homeomorphisms by Proposition 3.1, the projections from the pullback $\Sigma \times _{{\varphi },{\varphi }'}\Sigma '$ along ${\varphi },{\varphi }'$ to $\Sigma $ and $\Sigma '$ are local homeomorphisms, and hence the pullback is an ample groupoid. The Baer sum $\Sigma \oplus \Sigma '$ is the groupoid $\Sigma \times _{{\varphi },{\varphi }'}\Sigma '/A$ , where A acts on $\Sigma \times _{{\varphi },{\varphi }'}\Sigma '$ by $a(s,s') = (a^{-1} s, as')$ and we use the quotient topology. This a properly discontinuous action of a discrete group and so the quotient map $(\Sigma \times _{{\varphi },{\varphi }'}\Sigma ')\to \Sigma \oplus \Sigma '$ is a covering map, and hence a local homeomorphism. This equivalence relation is compatible with multiplication and so gives a groupoid structure which makes $\Sigma \oplus \Sigma '$ an ample groupoid. We leave the straightforward verifications to the interested reader. The inclusion $\kappa $ of $A\times {\mathscr G}^{(0)}$ into $\Sigma \oplus \Sigma '$ sends $(a,x)$ to $[x,ax]$ and the quotient map $\rho $ from $\Sigma \oplus \Sigma '$ to ${\mathscr G}$ sends $[s,s']$ to ${\varphi }(s)={\varphi }'(s')$ .

Theorem 3.20. The set of equivalence classes of A-twists over ${\mathscr G}$ under Baer sum is an abelian group isomorphic to $H^2(\Gamma _c({\mathscr G}),C_c({\mathscr G}^{(0)}, A\cup \{0\}))$ .

Proof. By Theorem 3.19, we just need to show that the cohomology class corresponding to the equivalence class of $\Sigma \oplus \Sigma '$ is the product of the cohomology classes corresponding to $\Sigma $ and $\Sigma '$ . Let $j\colon \Gamma _c({\mathscr G})\to \Gamma _c(\Sigma )$ and $j'\colon \Gamma _c({\mathscr G})\to \Gamma _c(\Sigma ')$ be set-theoretic sections with j and $j'$ preserving idempotents. Then the cohomology classes corresponding to $\Sigma $ and $\Sigma '$ are given by normalized $2$ -cycles $c,c'$ , respectively, where $j(U)j(V) = \iota (\gamma (c(U,V)))j(UV)$ and $j'(U)j'(V)=\iota '(\gamma (c'(U,V)))j'(UV)$ where $\gamma \colon \widetilde {A}\to \Gamma _c(A\times {\mathscr G}^{(0)})$ is the isomorphism of Proposition 3.16. Define $\widetilde {j}\colon \Gamma _c({\mathscr G})\to \Gamma _c(\Sigma \oplus \Sigma ')$ by

$$ \begin{align*} \widetilde{j}(U) = \{[g,g']\mid g\in j(U), g'\in j'(U)\}. \end{align*} $$

This is a compact open set since it is the image of $j(U)\times _{{\varphi },{\varphi }'} j'(U)$ , which is compact open in $\Sigma \times _{{\varphi },{\varphi }'}\Sigma '$ . It is a bisection because if ${\mathop {\boldsymbol d}}([g,g']) = {\mathop {\boldsymbol d}}([h,h'])$ for some $g,h\in j(U)$ and $g',h'\in j'(U)$ , then ${\mathop {\boldsymbol d}} (g)={\mathop {\boldsymbol d}}(h)$ and ${\mathop {\boldsymbol d}}(g')={\mathop {\boldsymbol d}}(h')$ , and so $g=g'$ and $h=h'$ . Similarly, ${\mathop {\boldsymbol r}}|_{\widetilde {j}(U)}$ is injective. By construction, $\rho ({\kern2pt}\widetilde {j}(U))=U$ . Also, if $U\subseteq {\mathscr G}^{(0)}$ , then $\widetilde {j}(U)\subseteq (\Sigma \oplus \Sigma ')^{(0)}$ . Let us compute the normalized $2$ -cocycle $\widetilde {c}$ corresponding to $\widetilde {j}$ .

Let $U,V\in \Gamma _c({\mathscr G})$ . Then $\widetilde {j}(U)\widetilde {j}(V)({\kern2pt}\widetilde {j}(UV))^*$ consists of all composable products $[g,g'][h,h'][k^{-1}, (k')^{-1}]$ , for $g\in j(U)$ , $h\in j(V)$ , $k\in j(UV)$ , $g'\in j'(U)$ , $h'\in j'(V)$ and $k'\in j'(UV)$ . Then since $U,V$ are bisections, it follows from composability that ${\varphi }(g){\varphi }(h)={\varphi }(k)$ and ${\varphi }'(g'){\varphi }'(h')={\varphi }'(k')$ . Therefore, $gh=ak$ and $g'h'=a'k'$ for some $a,a'\in A$ . But, using $j(U)j(V) = \iota (\gamma (c(U,V)))j(UV)$ and $j'(U)j'(V)=\iota '(\gamma (c'(U,V)))j'(UV)$ , we deduce that $a=c(U,V)({\mathop {\boldsymbol r}}(gh))$ and $a'=c'(U,V)({\mathop {\boldsymbol r}}(gh))$ . Thus

$$ \begin{align*} [g,g'][h,h'][k^{-1}, (k')^{-1}] &= [c(U,V)({\mathop{\boldsymbol r}}(gh)){\mathop{\boldsymbol r}}(gh),c'(U,V)({\mathop{\boldsymbol r}}(gh)){\mathop{\boldsymbol r}}(gh)]\\ &= [{\mathop{\boldsymbol r}}(gh),(c(U,V)c'(U,V))({\mathop{\boldsymbol r}}(gh)){\mathop{\boldsymbol r}}(gh)]\\ &= \kappa((c(U,V)c'(U,V))({\mathop{\boldsymbol r}}(gh)),{\mathop{\boldsymbol r}}(gh)). \end{align*} $$

It follows that $\widetilde {c}(U,V) = c(U,V)c'(U,V)$ , as required.

We remark that the easiest way to build an A-twist over ${\mathscr G}$ is to begin with a normalized locally constant $2$ -cocycle $c\colon {\mathscr G}^{(2)} \to A$ (where ${\mathscr G}^{(2)}$ is the space of composable pairs of elements of ${\mathscr G}$ ). Being a $2$ -cocycle means that $c(h,k)c(g,hk)=c(g,h)c(gh,k)$ , and being normalized means $c(x,x)=1$ for all units $x\in {\mathscr G}^{(0)}$ . From a normalized $2$ -cocycle, one can build a twist $\Sigma = A\times {\mathscr G}$ with the product topology, ${\mathop {\boldsymbol d}}(a,g)=(1,{\mathop {\boldsymbol d}}(g))$ , ${\mathop {\boldsymbol r}}(a,g)=(1,{\mathop {\boldsymbol r}}(g))$ and $(a,g)(b,h) = (abc(g,h),gh)$ . The inverse is given by $(a,g)^{-1} = (a^{-1} c(g,g^{-1})^{-1}, g^{-1})$ . The corresponding exact sequence is

$$ \begin{align*} A\times {\mathscr G}^{(0)}\to \Sigma\to {\mathscr G}, \end{align*} $$

where the first map is the inclusion and the second is the projection. One easily adapts [5, Proposition 4.8] to general A to show that an A-twist as in (3-3) is equivalent to one coming from a locally constant $2$ -cocycle on ${\mathscr G}$ if and only if there is a continuous section $s\colon {\mathscr G}\to \Sigma $ with $s({\mathscr G}^{(0)})\subseteq \Sigma ^{(0)}$ . By Proposition 3.8 this occurs if and only if $\Gamma _c({\varphi })\colon \Gamma _c(\Sigma )\to \Gamma _c({\mathscr G})$ admits an order-preserving and idempotent-preserving section. Proposition 3.9 recovers the well-known result that any twist over a second countable Hausdorff ample groupoid comes from a $2$ -cocycle [5], and extends it to the paracompact case.

4 Inverse semigroup crossed products

It was observed in [7] that Steinberg algebras of Hausdorff ample groupoids are skew inverse semigroup rings (see [14] for the non-Hausdorff case). We now define a notion of inverse semigroup crossed product that captures twisted Steinberg algebras. Even without a twist, our definition looks different from the definition of a skew inverse semigroup ring found in the literature [6, 7], but it coincides with that definition for the case of so-called spectral actions [18], which arises in the ample groupoid setting.

4.1 Actions

Let R be a ring (associative, but not necessarily unital or commutative). We say that a ring endomorphism $\psi \colon R\to R$ is proper if $\psi (R)=Re$ for some central idempotent e of R (necessarily unique). Note that if $e,f\in E(Z(R))$ are central idempotents, then $Re\cap Rf=Ref=ReRf$ .

Proposition 4.1. The proper ring endomorphisms of R form a semigroup ${\mathop {\mathrm {End}}}_c(R)$ under composition. The idempotent proper endomorphisms are those of the form ${\varphi }_e(r)=re$ for some $e\in E(Z(R))$ and, hence, $E({\mathop {\mathrm {End}}}_c(R))$ is a commutative subsemigroup isomorphic to the Boolean algebra $E(Z(R))$ of central idempotents of R (under meet). Moreover, ${\varphi }(Z(R))\subseteq Z(R)$ for any proper endomorphism ${\varphi }$ .

Proof. For ${\varphi }\in {\mathop {\mathrm {End}}}_c(R)$ , we put ${\varphi }(R)=Re_{{\varphi }}$ , where $e_{{\varphi }}\in E(Z(R))$ . First we claim that if ${\varphi }\in {\mathop {\mathrm {End}}}_c(R)$ and $z\in Z(R)$ , then ${\varphi }(z)\in Z(R)$ . Indeed, if $r\in R$ , then $re_{{\varphi }}\in Re_{{\varphi }}={\varphi }(R)$ and so $re_{{\varphi }} = {\varphi }(r')$ for some $r'\in R$ . Thus

$$ \begin{align*} {\varphi }(z)r &= ({\varphi }(z)e_{{\varphi }})r = {\varphi }(z)(re_{{\varphi }})={\varphi }(z){\varphi }(r') = {\varphi }(zr')={\varphi }(r'z) = {\varphi }(r'){\varphi }(z) \\ &= re_{{\varphi }}{\varphi }(z)=r{\varphi }(z). \end{align*} $$

So if ${\varphi },\psi \in {\mathop {\mathrm {End}}}_c(R)$ , then ${\varphi }(\psi (R)) = {\varphi }(Re_{\psi }) = {\varphi }(R){\varphi }(e_{\psi }) = Re_{{\varphi }}{\varphi }(e_{\psi })$ and $e_{{\varphi }}{\varphi }(e_{\psi })$ is a central idempotent as ${\varphi }(Z(R))\subseteq Z(R)$ .

Clearly, ${\varphi }_e(r)=re$ , where e is a central idempotent is a proper endomorphism with image $Re$ . Conversely, if ${\varphi }\in {\mathop {\mathrm {End}}}_c(R)$ is an idempotent, then ${\varphi }$ fixes ${\varphi }(R) = Re_{{\varphi }}$ . So if $r\in R$ , then ${\varphi }(r)={\varphi }(r)e_{{\varphi }}={\varphi }(r){\varphi }(e_{{\varphi }}) = {\varphi }(re_{{\varphi }}) =re_{{\varphi }}$ , as required. This completes the proof.

Since ${\mathop {\mathrm {End}}}_c(R)$ has commuting idempotents, the von Neumann regular elements of ${\mathop {\mathrm {End}}}_c(R)$ form an inverse semigroup. Thus it is natural to consider actions of inverse semigroups on rings by proper endomorphisms.

Definition 4.2. We define an action of an inverse semigroup S on a ring R to be a homomorphism $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ , written $s\mapsto \alpha _s$ . Often we write $sr$ for $\alpha _s(r)$ . If $e\in E(S)$ , let $1_e$ be the central idempotent with $\alpha _e(R) = R1_e$ . We say that an action is nondegenerate if $\alpha $ is idempotent separating and $R=\sum _{e\in E(S)}R1_e$ .

A set E of idempotents of a ring R is called a set of local units if, for all finite subsets F of R, there is an idempotent $e\in E$ with $F\subseteq eRe$ . If S admits a nondegenerate action on R, then it is easy to check that the Boolean algebra generated by the $1_e$ with $e\in E(S)$ is a set of local units for R. If S is a Boolean inverse semigroup and $\alpha $ is additive, then the set of $1_e$ with $e\in E(S)$ is a set of local units.

Proposition 4.3. Let S have a nondegenerate action $\alpha $ on R. Then

1. (1) $\alpha _s(R) = R1_{ss^*}$ ;

2. (2) $\alpha _s(R1_e) = R1_{ses^*}$ .

Proof. Now, $\alpha _s(R) = \alpha _{ss^*}(\alpha _s(R))\subseteq R1_{ss*}$ . On the other hand, since $\alpha _{ss^*}(r) = r1_{ss^*}$ by Proposition 4.1,

$$ \begin{align*}r1_{ss^*} = \alpha_{ss^*}(r) = \alpha_s(\alpha_{s^*}(r))\in \alpha_s(R).\end{align*} $$

This establishes the first item. The second follows from the first because $\alpha _s(R1_e) = \alpha _s\alpha _e(R)=\alpha _{se}(R)=R1_{ses^*}$ .

We typically are interested in the case where S has a zero and $\alpha $ preserves $0$ (so $\alpha _0(r)=0$ for all $r\in R$ ). We call such an action zero-preserving.

One should note that $\alpha _s$ restricts to an isomorphism $R1_{s^*s}\to R1_{ss^*}$ , as is easily checked.

To define a notion of crossed product, we next need to consider twists. Let S have a nondegenerate action on a ring R. Put $\widetilde {R} = \bigcup _{e\in E(S)}(Z(R)1_e)^{\times }$ .

Proposition 4.4. The set $\widetilde {R}$ is a commutative inverse semigroup under multiplication with $E(\widetilde {R}) =\{1_e\mid e\in E(S)\}$ . Moreover, if $s\in S$ , then $\alpha _s(\widetilde {R})\subseteq \widetilde {R}$ and S acts on $\widetilde {R}$ by endomorphisms.

Proof. Given $r,r'\in (Z(R)1_e)^\times $ and $u,u'\in (Z(R)1_f)^\times $ satisfying $rr'=1_e$ and $uu'=1_f$ , we have $ru,u'r'\in Z(R)1_{ef}$ and $ruu'r' = r1_fr'=rr'1_f=1_e1_f=1_{ef}$ . Thus $ru,u'r'\in \widetilde {R}$ and $(ru)(u'r')(ru)=ru$ . Therefore, $\widetilde {R}\subseteq Z(R)$ is a commutative von Neumann regular semigroup and, hence, inverse. Clearly, $E(\widetilde {R}) =\{1_e\mid e\in E(S)\}$ .

If $s\in S$ and $r,r'\in (Z(R)1_e)^\times $ satisfy $rr'=1_e$ , then $\alpha _s(r)\alpha _s(r') = \alpha _s(1_e) = 1_{ses^*}$ , and so $\alpha _s(r)\in (Z(R)1_{ses^*}{\kern-1pt})^\times $ (using Propositions 4.1 and 4.3).

We have now established that if the action is nondegenerate, then $\widetilde {R}$ is an S-module, where we put $p(r)=e$ if $r\in (Z(R)1_e)^\times $ by Proposition 4.3(2).

4.2 Crossed products

We continue to work in the context of an inverse semigroup S with a nondegenerate action $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ on R. Fix a normalized $2$ -cocycle $c\colon S\times S\to \widetilde {R}$ . To define the crossed product $R\rtimes _{\alpha ,c} S$ , we proceed in two steps.

Proposition 4.5. Let S be an inverse semigroup and let $\alpha $ be a nondegenerate action of S on a ring R. Let $c\colon S\times S\to \widetilde {R}$ be a normalized $2$ -cocycle.

1. (1) The abelian group $\bigoplus _{s\in S} R\delta _s$ (here $\delta _s$ is an indexing symbol) is a ring with product defined by $r\delta _s\cdot r'\delta _t = r(sr')c(s,t)\delta _{st}$ .

2. (2) The additive subgroup I generated by $r\delta _s-rc(t,s^*s)^*\delta _t$ with $s\leq t$ is a two-sided ideal. If S has a zero and the action is zero-preserving, then I contains $R\delta _0$ .

Proof. For the first item, it suffices to check associativity. We calculate

$$ \begin{align*} ((r_1\delta_s)(r_2\delta_t))r_3\delta_u &= r_1(sr_2)c(s,t)(str_3)c(st,u) \delta_{stu}, \\ r_1\delta_s(r_2\delta_t r_3\delta_u) &= r_1s(r_2(tr_3)c(t,u))c(s,tu)\delta_{stu}\\ &= r_1(sr_2)(str_3)(sc(t,u))c(s,tu)\delta_{stu}. \end{align*} $$

Since c takes values in the center of R, associativity follows from c being a $2$ -cocycle.

The second item is more technical. Consider $s\leq t\in S$ and $r\in R$ . If $u\in S$ and $a\in R$ , then

$$ \begin{align*}a\delta_u(r\delta_s-rc(t,s^*s)^*\delta_t)= a(ur)c(u,s)\delta_{su} - a(ur)(uc(t,s^*s)^*)c(u,t)\delta_{ut}.\end{align*} $$

Since $su\leq ut$ , it suffices to show that $c(u,s)c(ut, s^*u^*us)^* = (uc(t,s^*s)^*)c(u,t)$ , which is the content of Proposition 3.11(10). On the other hand,

$$ \begin{align*}(r\delta_s-rc(t,s^*s)^*\delta_t)a\delta_u = r(sa)c(s,u)\delta_{su}-rc(t,s^*s)^*(ta)c(t,u)\delta_{tu}.\end{align*} $$

Note that $c(t,s^*s)\in R1_{ss^*}$ and so

$$ \begin{align*}c(t,s^*s)^*(ta) = c(t,s^*s)^*1_{ss^*}(ta) = c(t,s^*s)^*(ta)1_{ss^*} = c(t,s^*s)^*(ss^*(ta)) = c(t,s^*s)^*(sa).\end{align*} $$

Thus it suffices to show that $r(sa)c(s,u)\delta _{su}-r(sa)c(t,s^*s)^*c(t,u)\delta _{tu}\in I$ . Since $su\leq tu$ , it suffices to show that $c(s,u)c(tu,u^*s^*su) = c(t,s^*s)^*c(t,u)$ , which follows from Proposition 3.11(11). This completes the proof that I is an ideal.

If $\alpha $ is zero-preserving, then, since $0\leq 0$ , we have $r\delta _0-rc(0,0)^*\delta _0\in I$ . But $c(0,0)=0$ as $1_0=0$ and so $(Z(R)1_0)^\times =0$ . Thus $r\delta _0\in I$ .

We can now define the crossed product.

Definition 4.6. Let R be a ring, S an inverse semigroup and $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ a nondegenerate action. Let $c\colon S\times S\to \widetilde {R}$ be a normalized $2$ -cocycle. Then the crossed product $R\rtimes _{\alpha ,c} S$ is the ring $\bigoplus _{s\in S}R\delta _s/I$ , where we retain the notation of Proposition 4.5.

It is relatively straightforward to verify that two cohomologous normalized $2$ -cocycles yield isomorphic crossed products.

Proposition 4.7. Let $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ be a nondegenerate action of S on R. Let $c,c'\colon S\times S\to \widetilde {R}$ be normalized $2$ -cocycles. If c and $c'$ are cohomologous, then $R\rtimes _{\alpha ,c} S\cong R\rtimes _{\alpha ,c'} S$ .

Proof. Let $F\colon S\to \widetilde {R}$ be a mapping with $F(s)\in (Z(R)1_{ss^*}{\kern-1pt})^\times $ , for all $s\in S$ , and with $c=c'\cdot \delta F$ . Then $c(s,t) = c'(s,t)F(s)(sF(t))F(st)^*$ . Since $c,c'$ are normalized, $c(e,e)=1_e=c'(e,e)$ for $e\in E(S)$ . Thus

$$ \begin{align*} F(e)= c'(e,e)F(e)(eF(e))F(ee)^* = c(e,e)=1_e \end{align*} $$

and F is idempotent-preserving.

We put $A=\bigoplus _{s\in S}R\delta _s$ with product $r\delta _s\cdot r'\delta _t = r(sr')c(s,t)\delta _{st}$ and let I be the additive subgroup generated by all $r\delta _s-rc(t,s^*s)^*\delta _t$ with $s\leq t$ . Similarly, we let $A'=\bigoplus _{s\in S}R\delta _s$ with multiplication $r\delta _s\cdot r'\delta _t = r(sr')c'(s,t)\delta _{st}$ and $I'$ be the additive subgroup generated by all $r\delta _s-rc'(t,s^*s)^*\delta _t$ with $s\leq t$ . We first construct a homomorphism $\Phi \colon A\to A'$ with $\Phi (I)\subseteq I'$ . Define $\Phi (r\delta _s) = rF(s)\delta _s$ . Note that $\Phi (r\delta _s\cdot r'\delta _t) = r(sr')c(s,t)F(st)\delta _{st}$ , whereas $\Phi (r\delta _s)\Phi (r'\delta _t) = rF(s)\delta _s\cdot r'F(t)\delta _t = r(sr')F(s)(sF(t))c'(s,t)\delta _{st}$ . But $F(st)^*F(st) = 1_{stt^*s^*}$ and $c'(s,t)\in R1_{stt^*s^*}$ , whence

$$ \begin{align*} c'(s,t) = F(st)^*F(st)c'(s,t). \end{align*} $$

Therefore,

$$ \begin{align*} \Phi(r\delta_s)\Phi(r'\delta_s) &= r(sr')F(s)(sF(t))F(st)^*c'(s,t)F(st)\delta_{st} \\ &= r(sr')c(s,t)F(st)\delta_{st}=\Phi(r\delta_s\cdot r'\delta_t). \end{align*} $$

We conclude that $\Phi $ is a homomorphism.

Suppose now that $s\leq t$ and note that

$$ \begin{align*} \Phi (r\delta _s-rc(t,s^*s)^*\delta _t)\! = rF(s)\delta _s\! - rc(t,s^*s)^* F(t)\delta _t. \end{align*} $$

But

$$ \begin{align*} c(t,s^*s)^*=c'(t,s^*s)^*F(t)^*(tF(s^*s))^*F(ts^*s) = c'(t,s^*s)^*F(t)^*1_{ss^*}F(s) \end{align*} $$

as $F(s^*s) =1_{s^*s}$ and $ts^*s=s$ . Thus, using $1_{tt^*}\geq 1_{ss^*}$ , we have

$$ \begin{align*} rc(t,s^*s)^*F(t)=rF(s)c'(t,s^*s)^*F(t)^*F(t)1_{ss^*} = rF(s)c'(t,s^*s)^*. \end{align*} $$

Therefore, $ rF(s)\delta _s - rc(t,s^*s)^*F(t)\delta _t=rF(s)\delta _s-rF(s)c'(t,s^*s)^*\delta _t\in I'$ , and so we have a well-defined homomorphism ${\varphi }\colon R\rtimes _{\alpha ,c}S\to R\rtimes _{\alpha ,c'} S$ given by

$$ \begin{align*} {\varphi }(r\delta _s+I) = \Phi (r\delta _s)+I' = rF(s)\delta _s+I'. \end{align*} $$

Similarly, we have a well-defined homomorphism $\psi \colon R\rtimes _{\alpha ,c'}S\to R\rtimes _{\alpha ,c} S$ given by $\psi (r\delta _s+I') = rF(s)^*\delta _s+I$ . We show that these two homomorphisms are inverse to each other. By symmetry, it suffices to consider $\psi \circ {\varphi }$ . Now, $\psi {\varphi }(r\delta _s+I) = \psi (rF(s)\delta _s+I') = rF(s)F(s)^*\delta _s+I = r1_{ss^*}\delta _s+I$ . But since $s\leq s$ and $c(s,s^*s)= 1_{ss^*}$ by Proposition 3.11(1), it follows that $r\delta _s-r1_{ss^*}\delta _s = r\delta _s-rc(s,s^*s)^*\delta _s\in I$ . Thus $\psi {\varphi }(r\delta _s+I) = r\delta _s+I$ . This completes the proof.

When the $2$ -cocycle c is trivial, $R\rtimes _{\alpha ,c} S$ is called a skew inverse semigroup ring. Although it looks superficially different, this notion of a skew inverse semigroup ring coincides with the standard one for what are called spectral actions in [18]. Note that, when c is trivial, I is generated by all differences $r\delta _s-r1_{ss^*}\delta _t$ with $s\leq t$ .

The following two results yield a partial generalization of a result for skew inverse semigroup rings [6, Proposition 3.1].

Proposition 4.8. Let $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ be a nondegenerate action and $c\colon S\times S\to \widetilde {R}$ a normalized $2$ -cocycle. Then the additive subgroup of the ring $R\rtimes _{\alpha ,c} S$ generated by the elements $r\delta _e+I$ with $r\in R$ and $e\in E(S)$ is a subring isomorphic to a quotient of R. If there is an additive group homomorphism $\tau \colon R\rtimes _{\alpha ,c}S\to R$ with $\tau (r\delta _e+I) = r$ for all $r\in R1_e$ and $e\in E(S)$ , then this subring is isomorphic to R.

Proof. Let A be the additive subgroup generated by the $r\delta _e+I$ with $r\in R$ and $e\in E(S)$ . Since $r\delta _e\cdot r'\delta _f+I = r(er')c(e,f)\delta _{ef}+I$ , it is clear that R is a subring. Suppose that $r\in R$ . Since the action is nondegenerate, we may write $r=\sum _{i=1}^n r_i$ with $r_i\in R1_{e_i}$ and $e_i\in E(S)$ for $i=1,\ldots , n$ . Define $\rho (r) = \sum _{i=1}^n r_i\delta _{e_i}+I$ . We claim that $\rho $ is a well-defined homomorphism from R to A. To show that $\rho $ is well defined, it suffices to show that if $0=\sum _{i=1}^n r_i$ with $r_i\in R1_{e_i}$ , then $\sum _{i=1}^nr_i\delta _{e_i}\in I$ . We proceed by induction on n, with the case $n=1$ being trivial, as then $r_1=0$ and so $r_1\delta _{e_1}=0\in I$ . Suppose now that the result is true for $n-1$ and $0=\sum _{i=1}^n r_i$ with $r_i\in R1_{e_i}$ and $n\geq 2$ . Then $r_n = -\sum _{i=1}^{n-1}r_i$ and so $r_n=r_n1_{e_n} = -\sum _{i=1}^{n-1}r_i1_{e_n}$ . Note that $r_i-r_i1_{e_n}\in R1_{e_i}$ and $0=\sum _{i=1}^{n-1} (r_i-r_i1_{e_n})$ . By induction, we deduce that $\sum _{i=1}^{n-1} (r_i-r_i1_{e_n})\delta _{e_i}\in I$ , that is,

(4-1) $$ \begin{align} \sum_{i=1}^{n-1}r_i\delta_{e_i}+I = \sum_{i=1}^{n-1}r_i1_{e_n}\delta_{e_i}+I. \end{align} $$

Since $e_ie_n\leq e_i$ , we have $r_i\delta _{e_ie_n}-r_i1_{e_n}\delta _{e_i}\in I$ , as $r_ic(e_i, e_ie_n)^* = r_i1_{e_i}1_{e_n}=r_i1_{e_n}$ by Proposition 3.11(4). On the other hand, since $e_ie_n\leq e_n$ , we have $r_i\delta _{e_ie_n} - r_i1_{e_n}\delta _{e_n}\in I$ as $r_ic(e_n,e_ie_n)^* = r_i1_{e_i}1_{e_n} = r_i1_{e_n}$ . Therefore, $r_i1_{e_n}\delta _{e_i}+I=r_i1_{e_n}\delta _{e_n}+I$ . Thus, we deduce from (4-1) that

$$ \begin{align*} \sum_{i=1}^{n-1}r_i\delta_{e_i}+I = \sum_{i=1}^{n-1}r_i1_{e_n}\delta_{e_n}+I = -r_n\delta_{e_n}+I. \end{align*} $$

It follows that $\sum _{i=1}^nr_i\delta _{e_i}\in I$ , as required.

To verify that $\rho $ is a homomorphism, let $r=\sum _{i=1}^n r_i$ and $r'=\sum _{j=1}^m r^{\prime }_j$ , where $r_i\in R1_{e_i}$ and $r_j\in R1_{f_j}$ with $e_i, f_j\in E(S)$ for $1\leq i\leq n$ , $1\leq j\leq m$ . Then $rr' = \sum _{i=1}^n\sum _{j=1}^m r_ir^{\prime }_j$ and $r_ir^{\prime }_j\in R1_{e_if_j}$ . Therefore, $\rho (rr') = \sum _{i=1}^n\sum _{j=1}^m r_ir^{\prime }_j\delta _{e_if_j}+I$ . On the other hand,

$$ \begin{align*} \rho(r)\rho(r') =\sum_{i=1}^nr_i\delta_{e_i}\sum_{j=1}^mr_j\delta_{f_j}+I=\sum_{i=1}^n\sum_{j=1}^m r_i(e_ir_j)c(e_i,f_j)\delta_{e_if_j}+I. \end{align*} $$

But $e_ir_j = r_j1_{e_i}$ and $c(e_i,f_j) =1_{e_if_j}$ by Proposition 3.11(4) and so

$$ \begin{align*} r_i(e_ir_j)c(e_i,f_j)=r_ir_j1_{e_i}1_{e_if_j} = r_ir_j. \end{align*} $$

Therefore, $\rho $ is a homomorphism.

To check $\rho $ is surjective, notice that, since $e\leq e$ , we have $r\delta _e+I = rc(e,e)^*\delta _e+I = r1_e\delta _e+I = \rho (r1_e)$ . Suppose that $\tau $ , as in the final statement, exists. If $r\in R$ with $r=\sum _{i=1}^n r_i$ and $r_i\in R1_{e_i}$ , for $i=1,\ldots , n$ , then $\tau ({\kern1pt}\rho (r)) = \tau (\sum _{i=1}^n r_i\delta _{e_i}+I)=\sum _{i=1}^n r_i = r$ and so $\rho $ is injective.

We suspect that $\rho $ is injective in general, but we could only prove the existence of the additive homomorphism $\tau $ in certain cases.

Proposition 4.9. Let $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ be a nondegenerate action and $c\colon S\times S\to \widetilde {R}$ a normalized $2$ -cocycle. Then an additive group homomorphism $\tau \colon R\rtimes _{\alpha ,c}S\to R$ satisfying $\tau (r\delta _e+I) = r$ for all $r\in R1_e$ exists in the following cases:

1. (1) if $c(t,s^*s)=1_{ss^*}$ whenever $s\leq t$ (for example, if c is trivial, that is, $R\rtimes _{\alpha ,c} S$ is a skew inverse semigroup ring);

2. (2) if, for each $s\in S$ , the set $E(s)=\{e\in E(S)\mid e\leq s\}$ either is empty or has a unique maximum element $e(s)$ .

Proof. Suppose the first condition is satisfied. Define a homomorphism of additive groups $\tau '\colon \bigoplus _{s\in S}R\delta _s\to R$ by $r\delta _s\mapsto r1_{ss^*}$ . Then a generator for I is of the form $r\delta _s-r1_{ss^*}\delta _t$ with $s\leq t$ , as $c(t,s^*s)^*=1_{ss^*}$ , and $\tau '(r\delta _s-r1_{ss^*}\delta _t) = r1_{ss^*}-r1_{ss^*}1_{tt^*} =0$ since $s\leq t$ . Therefore, $\tau '$ induces an additive group homomorphism $\tau \colon R\rtimes _{\alpha ,c}S\to R$ . Moreover, $\tau (r\delta _e+I) = r1_e=r$ for $r\in R1_e$ , where $e\in E(S)$ .

Now, suppose that each set $E(s)$ either is empty or has a maximum $e(s)$ . Define a homomorphism of additive groups $\tau '\colon \bigoplus _{s\in S}R\delta _s\to R$ by

$$ \begin{align*} \tau'(r\delta_s) = \begin{cases} rc(s,e(s)) & \text{if } E(s)\neq \emptyset , \\[-2pt] 0 & \text{otherwise.}\end{cases} \end{align*} $$

We need to check that $I\subseteq \ker \tau '$ . Suppose that $s\leq t$ . Obviously $E(s)\subseteq E(t)$ . On the other hand, if $e\in E(t)$ , then $es^*s\leq ts^*s=s$ and so $es^*s\in E(s)$ . Thus $E(s)=\emptyset $ if and only if $E(t)=\emptyset $ . In particular, if $s\leq t$ and $E(s)=\emptyset =E(t)$ , then ${\tau '(s\delta _s-sc(t,s^*s)^*\delta _t) =0}$ . So let us assume that $E(s)$ and $E(t)$ are nonempty. Then $e(s)\leq s\leq t$ implies $e(s)\leq e(t)$ and $e(s)\leq s^*s$ . Thus $e(s)\leq e(t)s^*s$ . But $e(t)s^*s\leq ts^*s=s$ , and so $e(t)s^*s\leq e(s)$ . Therefore, $e(t)s^*s=e(s)$ . A similar argument shows that $ss^*e(t)=e(s)$ .

We now compute $(tc(s^*s,e(s)))c(t,s^*se(s)) = c(t,s^*s)c(ts^*s,e(s))$ , which yields $c(t,e(s))=c(t,s^*s)c(s,e(s))$ since c is normalized (using Proposition 3.11(4)). But

$$ \begin{align*} (tc(e(t),s^*s))c(t,e(t)s^*s)=c(t,e(t))c(te(t),s^*s) = c(t,e(t))c(e(t),s^*s). \end{align*} $$

As $e(t)s^*s=e(s)$ , we deduce, using Proposition 3.11(4), that $c(t,e(s))=c(t,e(t))1_{e(s)}=e(s)c(t,e(t))$ . Therefore, $e(s)c(t,e(t))=c(t,s^*s)c(s,e(s))$ . Also note that $p(c(t,e(t))1_{ss^*}{\kern-1pt}) = ss^*e(t)=e(s)$ and so

$$ \begin{align*} c(t,e(t))1_{ss^*}=e(s)(c(t,e(t))1_{ss^*}{\kern-1pt}) =e(s)(ss^*c(t,e(t))) = e(s) c(t,e(t))= c(t,s^*s)c(s,e(s)). \end{align*} $$

Thus

$$ \begin{align*} \tau'(r\delta_s-rc(t,s^*s)^*\delta_t)& = rc(e,s)-rc(t,s^*s)^*c(t,e(t))\\[-2pt] &= rc(e,s)-rc(t,s^*s)^*1_{ss^*}c(t,e(t)) \\[-2pt] &= rc(e,s)-rc(t,s^*s)^*c(t,s^*s)c(s,e(s))\\[-2pt] &=0 \end{align*} $$

as $ss^*\geq e(s)$ . Therefore, because c is normalized, $\tau '$ induces a well-defined map $\tau \colon R\rtimes _{\alpha ,c}S\to R$ with $\tau (r\delta _e+I)=\tau '(r\delta _e) =rc(e,e)=r1_e=r$ if $r\in R1_e$ .

In addition to the case where c is trivial, we note that if (3-1) is an extension of inverse semigroups that admits an order-preserving and idempotent section $j\colon S\to T$ , then the corresponding normalized $2$ -cocycle c satisfies $c(t,s^*s)=1_{ss^*}$ when $s\leq t$ since $j(t)j(s^*s)=j(ts^*s)=j(s)$ by Lemma 3.7. Recall that an inverse semigroup S is E-unitary if $s\geq e$ with $e\in E(S)$ implies $s\in E(S)$ , and S is $0$ -E-unitary if S has a zero and $s\geq e\neq 0$ with $e\in E(S)$ implies $s\in E(S)$ ; see [21] for details. Examples of inverse semigroups satisfying the second condition of Proposition 4.9 include E-unitary and $0$ -E-unitary inverse semigroups and the inverse semigroup of compact open bisections of a Hausdorff ample groupoid ${\mathscr G}$ .

We now establish a universal property for the crossed product, generalizing that of skew inverse semigroup rings [18]. Let S be an inverse semigroup with a nondegenerate action $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ on a ring R and let $c\colon S\times S\to \widetilde {R}$ be a normalized $2$ -cocycle. Then a covariant representation of $(\alpha ,c)$ in a ring A consists of a ring homomorphism $\rho \colon R\to A$ and a map $\psi \colon S\to A$ such that:

1. (C1) $\psi (s)\rho (r) = \rho (sr)\psi (s)$ ;

2. (C2) $\rho (1_e) = \psi (e)$ ;

3. (C3) $\psi (s)\psi (t) = \rho (c(s,t))\psi (st)$ ; and

4. (C4) $\psi (ss^*)\psi (s) =\psi (s)$

for all $s,t\in S$ , $r\in R$ and $e\in E(S)$ . Note that (C4) is equivalent to

1. (C4^′) $\psi (s)\psi (s^*s)=\psi (s)$

in the presence of (C1), (C2). Indeed,

$$ \begin{align*} \psi(s)\psi(s^*s)=\psi(s)\rho(1_{s^*s})=\rho(1_{ss^*}{\kern-1pt})\psi(s)=\psi(ss^*)\psi(s)=\psi(s) \end{align*} $$

by (C1), (C2) and (C4). So (C4) implies (C4^′) and the reverse implication is dual. In the case where c is trivial, the conjunction of (C3) and (C4) is equivalent to $\psi $ being a semigroup homomorphism, and so the notion of covariant representation for skew inverse semigroup rings given here coincides with that considered in [18].

Proposition 4.10. Let S be an inverse semigroup with a nondegenerate action $\alpha \colon S{\kern-1pt}\to{\kern-1pt}{\mathop {\mathrm {End}}}_c(R)$ on a ring R and $c\colon S\times S{\kern-1pt}\to{\kern-1pt}\widetilde {R}$ a normalized $2$ -cocycle. Let $\rho \colon R{\kern-1pt}\to{\kern-1pt}R\rtimes _{\alpha ,c} S$ be the homomorphism from the proof of Proposition 4.8 and let $\psi \colon S{\kern-1pt}\to{\kern-1pt}R\rtimes _{\alpha ,c} S$ be given by $\psi (s) = 1_{ss^*}\delta _s+I$ . Then $({\kern1pt}\rho ,\psi )$ is a covariant representation and it is the universal covariant representation; that is, if $\rho '\colon R{\kern-1pt}\to{\kern-1pt}A$ and $\psi '\colon S{\kern-1pt}\to{\kern-1pt}A$ give a covariant representation, then there is a unique homomorphism $\pi \colon R\rtimes _{\alpha ,c}S{\kern-1pt}\to{\kern-1pt}A$ such that $\rho '=\pi \rho $ and $\psi '=\pi \psi $ . Namely, one has $\pi (r\delta _s+I) = \rho '(r)\psi '(s)$ .

Proof. We first check that $\rho ,\psi $ yield a covariant representation. If $r\in R$ satisfies $r=\sum _{i=1}^nr_i$ , where $r_i\in R1_{e_i}$ , then $\rho (r)=\sum _{i=1}^nr_i\delta _{e_i}+I$ and so

(4-2) $$ \begin{align} \psi(s)\rho(r) = \sum_{i=1}^n1_{ss^*}\delta_s \cdot r_i\delta_{e_i}+I = \sum_{i=1}^n(sr_i)c(s,e_i)\delta_{se_i}+I. \end{align} $$

On the other hand, $sr = \sum _{i=1}^nsr_i$ and $sr_i\in R1_{se_is^*}$ . Therefore,

(4-3) $$ \begin{align} \rho(sr)\psi(s) = \sum_{i=1}^nsr_i\delta_{se_is^*}\cdot 1_{ss^*}\delta_s+I = \sum_{i=1}^n (sr_i)c(se_is^*,s)\delta_{se_i}+I. \end{align} $$

It follows that the right-hand sides of (4-2) and (4-3) are equal by Proposition 3.11(9), yielding (C1). For (C2), note that $\rho (1_e) = 1_e\delta _e+I = \psi (e)$ by definition. For (C3), we compute

$$ \begin{align*} \psi(s)\psi(t) &= 1_{ss^*}\delta_s\cdot 1_{tt^*}\delta_t+I = 1_{ss^*}(s1_{tt^*}{\kern-1pt})c(s,t)\delta_{st}+I\\ &= c(s,t)1_{stt^*s^*}\delta_{st}+I= c(s,t)\delta_{stt^*s^*}\cdot 1_{stt^*s^*}\delta_{st}+I\\ &= \rho(c(s,t))\psi(st) \end{align*} $$

since $c(s,t)\in R1_{stt^*s^*}$ . Here we use Proposition 4.3 to obtain $s(1_{tt^*}{\kern-1pt}) = 1_{stt^*s^*}$ and Proposition 3.11(1) to get $c(stt^*s^*,st)=1_{stt^*s^*}$ . This verifies (C3). Finally, (C4) follows because $\psi (ss^*)\psi (s) = 1_{ss^*}\delta _{ss^*}\cdot 1_{ss^*}\delta _s +I= 1_{ss^*}\delta _s+I$ by Proposition 3.11(1).

Suppose now that $\rho '\colon R\to A$ and $\psi '\colon S\to A$ give rise to a covariant representation. First we check that $r\delta _s\mapsto \rho '(r)\psi '(s)$ yields a well-defined homomorphism $\pi '\colon \bigoplus _{s\in S}R\delta _s\to A$ . Indeed,

$$ \begin{align*} \pi'(r_1\delta_{s_1})\pi'(r_2\delta_{s_2}) & = \rho'(r_1)\psi'(s_1)\rho'(r_2)\psi'(s_2) = \rho'(r_1)\rho'(s_1r_2)\psi'(s_1)\psi'(s_2)\\ &= \rho'(r_1(s_1r_2)c(s_1,s_2))\psi'(s_1s_2) = \pi'(r_1\delta_{s_1}\cdot r_2\delta_{s_2}) \end{align*} $$

by (C1) and (C3).

We now check that $I\subseteq \ker \pi '$ . Let $s\leq t$ and $r\in R$ . First note that $ts^*st^* = ss^*$ and so $c(t, s^*s)=c(ss^*,t)=c(ss^*,t)1_{ss^*}$ by Proposition 3.11(9). Therefore,

$$ \begin{align*} \pi'(rc(t,s^*s)^*\delta_t) &= \rho'(rc(ss^*,t)^*1_{ss^*}{\kern-1pt})\psi'(t) = \rho'(rc(ss^*,t)^*)\rho'(1_{ss^*}{\kern-1pt})\psi'(t)\\ & = \rho'(rc(ss^*,t)^*)\psi'(ss^*)\psi'(t)=\rho'(rc(ss^*,t)^*c(ss^*,t))\psi'(s)\\ &=\rho'(r1_{ss^*}{\kern-1pt})\psi'(s) = \rho'(r)\rho'(1_{ss^*}{\kern-1pt})\psi'(s)\\ &=\rho'(r)\psi'(ss^*)\psi'(s)= \rho'(r)\psi'(s)\\ &= \pi'(r\delta_s) \end{align*} $$

by (C2), (C3) and (C4). Thus $I\subseteq \ker \pi '$ and $\pi \colon R\rtimes _{\alpha ,c} S\to A$ given by $\pi (r\delta _s+I) = \rho '(r)\psi '(s)$ is well defined. If $r\in R$ satisfies $r=\sum _{i=1}^n r_i$ , where $r_i\in R1_{e_i}$ , then $\pi ({\kern1pt}\rho (r)) = \sum _{i=1}^n\pi '(r_i\delta _{e_i}) =\sum _{i=1}^n\rho '(r_i)\psi '(e_i) = \sum _{i=1}^n\rho '(r_i)\rho '(1_{e_i}) =\rho '(r)$ using (C2). Also

$$ \begin{align*}\pi(\psi(s)) = \pi'(1_{ss^*}\delta_s) = \rho'(1_{ss^*}{\kern-1pt})\psi'(s) = \psi'(ss^*)\psi'(s)=\psi'(s)\end{align*} $$

using (C2) and (C4).

For uniqueness, if $\gamma \colon R\times _{\alpha ,c} S\to A$ is a homomorphism satisfying $\gamma \rho =\rho '$ and $\gamma \psi =\psi '$ , then, since $r\delta _s+I = r1_{ss^*}\delta _s+I$ (as $s\leq s$ and $c(s,s^*s)=1_{ss^*}=c(ss^*,s)$ by Proposition 3.11(1)),

$$ \begin{align*} \gamma(r\delta_s+I) &= \gamma(r1_{ss^*}\delta_s+I) = \gamma(r1_{ss^*}\delta_{ss^*}\cdot 1_{ss^*}\delta_s+I)\\ &=\gamma({\kern1pt}\rho(r1_{ss^*}{\kern-1pt}))\gamma(\psi(s)) = \rho'(r1_{ss^*}{\kern-1pt})\psi'(s)\\ &=\rho'(r)\psi'(ss^*)\psi'(s)=\rho'(r)\psi'(s) \end{align*} $$

by (C2) and (C4), as required. This completes the proof.

We note that our notion of crossed product seems related to, but different from, the $C^*$ -algebraic twisted action crossed product in [12]. In [12], the action of an element of the inverse semigroup on a ring is only partially defined. Also, they loosen the requirement on how the action of a product of semigroup elements behaves. We could also allow greater generality by letting $\alpha \colon S\to {\mathop {\mathrm {End}}}_c(R)$ just be a map, not a homomorphism, and allowing the $2$ -cocycle c to take values in the noncommutative inverse semigroup $\bigcup _{e\in E(S)}(R1_e)^\times $ . However, the conditions on $\alpha $ and c that yield associativity are slightly more complicated. We do not need this more general construction here in any event as we are primarily interested in the case where R is commutative, in which case the general construction reduces to ours.

4.3 Twisted Steinberg algebras

Our goal is to show that twisted Steinberg algebras of Hausdorff ample groupoids are, in fact, inverse semigroup crossed products, generalizing the result of [7] for the case of untwisted Steinberg algebras and skew inverse semigroup rings. Let ${\mathscr G}$ be a Hausdorff ample groupoid and let

$$ \begin{align*} R^\times \times {\mathscr G}^{(0)}\xrightarrow{\,\,\iota\,\,}\Sigma\xrightarrow{\,\,{\varphi}\,\,} {\mathscr G} \end{align*} $$

be a discrete $R^\times $ -twist. The associated twisted Steinberg algebra $A_R({\mathscr G};\Sigma )$ consists of all locally constant functions $f\colon \Sigma \to R$ that are $R^\times $ -anti-equivariant, in the sense that $f(rs) = r^{-1} f(s)$ for $s\in \Sigma $ and $r\in R^\times $ , and have compact support modulo $R^\times $ , that is, ${\varphi }({\mathop {\mathrm {supp}}}(f))$ is compact. Addition and the R-module structure are pointwise. The product is defined as follows. Choose a set-theoretic section $p\colon {\mathscr G}\to \Sigma $ (we do not assume continuity of p). Then the convolution is given by

$$ \begin{align*} f\ast f'(s) = \sum_{{\mathop{\boldsymbol r}}(g){\kern1pt}={\kern1pt}{\mathop{\boldsymbol r}}({\varphi}(s))}f(p(g))f'(p(g)^{-1} s). \end{align*} $$

This does not depend on the choice of p. Note that we are following the conventions of [4] and not [5], which uses equivariant functions.

Let $A=C_c({\mathscr G}^{(0)},R)$ be the ring of compactly supported locally constant mappings ${\mathscr G}^{(0)}\to R$ with pointwise operations. There is a natural zero-preserving action $\alpha \colon \Gamma _c({\mathscr G})\to {\mathop {\mathrm {End}}}_c(A)$ given by

$$ \begin{align*} \alpha_U(f)(x) = \begin{cases} f({\mathop{\boldsymbol d}}(({\mathop{\boldsymbol r}}|_U)^{-1}(x))) & \text{if } x\in {\mathop{\boldsymbol r}}(U),\\ 0 & \text{otherwise.}\end{cases} \end{align*} $$

Notice that the image of $\alpha _U$ is $A1_{{\mathop {\boldsymbol r}}(U)}$ and that the action is nondegenerate and additive. Moreover, $\widetilde {A} = C_c({\mathscr G}^{(0)},R^\times \cup \{0\})$ with the action we previously considered in Proposition 3.15. (That $\alpha $ is an action is essentially the same proof as in Proposition 3.15.) Hence, we can consider a normalized $2$ -cocycle c associated to the twist as per Theorem 3.19 and form the crossed product $A\rtimes _{\alpha ,c} \Gamma _c({\mathscr G})$ , and the crossed product ring is independent of the choice of $2$ -cocycle up to isomorphism. More precisely, we fix a set-theoretic section $j\colon \Gamma _c({\mathscr G})\to \Gamma _c(\Sigma )$ with $j|_{E(\Gamma _c({\mathscr G}))} = (\Gamma _c({\varphi })|_{E(\Gamma _c(\Sigma ))})^{-1}$ . One then has

$$ \begin{align*} j(U)j(V) &= \iota(\{(c(U,V)(x),x)\mid x\in {\mathop{\boldsymbol r}}(UV)\})j(UV) \\ &= \{c(U,V)({\mathop{\boldsymbol r}}({\varphi}(s)))s\mid s\in j(UV)\}. \end{align*} $$

Our goal is to show that the crossed product $A\rtimes _{\alpha ,c} \Gamma _c({\mathscr G})$ is isomorphic to $A_R({\mathscr G};\Sigma )$ . Note that $\bigoplus _{U\in \Gamma _c({\mathscr G})}A\delta _U$ and $A\rtimes _{\alpha ,c} \Gamma _c({\mathscr G})$ are R-algebras with $r(a\delta _U) = (ra)\delta _U$ and the corresponding induced R-algebra structure on the crossed product, as is easily checked. Note that Propositions 4.8 and 4.9 apply in this setting to embed A into the crossed product since ${\mathscr G}$ is Hausdorff.

We proceed by first establishing a number of lemmas. We retain the notation I for the two-sided ideal from Proposition 4.5(2).

Lemma 4.11. Let $a\in A$ and $U\in \Gamma _c({\mathscr G})$ . Let $V={\mathop {\mathrm {supp}}}(a)\cap {\mathop {\boldsymbol r}}(U)$ . Then $a\delta _U+I = ac(U,{\mathop {\boldsymbol d}}(VU))\delta _{VU} +I$ and ${\mathop {\mathrm {supp}}}(ac(U,{\mathop {\boldsymbol d}}(VU))={\mathop {\boldsymbol r}}(VU)$ .

Proof. Now, $U\leq U$ and so $a\delta _U+I = ac(U,{\mathop {\boldsymbol d}}(U))^*\delta _U+I = a1_{{\mathop {\boldsymbol r}}(U)}\delta _U+I$ by Proposition 3.11(1), as c is normalized. Then $a1_{{\mathop {\boldsymbol r}}(U)} = a1_V=a1_{{\mathop {\boldsymbol r}}(VU)}$ by definition of V. Since $VU\leq U$ we have

$$ \begin{align*} ac(U,{\mathop{\boldsymbol d}}(VU))\delta_{VU}+I &= ac(U,{\mathop{\boldsymbol d}}(VU))c(U,{\mathop{\boldsymbol d}}(VU))^*\delta_{U}+I \\ &= a1_{{\mathop{\boldsymbol r}}(VU)}\delta_U+I= a\delta_U+I, \end{align*} $$

by the first line of the proof. Since ${\mathop {\mathrm {supp}}}(c(U,{\mathop {\boldsymbol d}}(VU)))={\mathop {\boldsymbol r}}(VU) \subseteq {\mathop {\mathrm {supp}}}(a)$ , we have that ${\mathop {\mathrm {supp}}} (ac(U,{\mathop {\boldsymbol d}}(VU)))={\mathop {\boldsymbol r}}(VU)$ .

Thus $A\rtimes _{\alpha ,c}\Gamma _c({\mathscr G})$ is spanned by elements of the form $a\delta _U+I$ with ${\mathop {\mathrm {supp}}}(a)={\mathop {\boldsymbol r}}(U)$ .

Lemma 4.12. Suppose that $U\in \Gamma _c({\mathscr G})$ is a union of pairwise disjoint compact open bisections $U_1,\ldots , U_k$ and ${\mathop {\mathrm {supp}}}(a)={\mathop {\boldsymbol r}}(U)$ . Then $a\delta _U+I = \sum _{i=1}^k a_i\delta _{U_i}+I$ with ${\mathop {\mathrm {supp}}}(a_i) = {\mathop {\boldsymbol r}}(U_i)$ for $i=1,\ldots , k$ .

Proof. Let $a_i = ac(U,{\mathop {\boldsymbol d}}(U_i))$ . Then ${\mathop {\mathrm {supp}}}(a_i)={\mathop {\mathrm {supp}}}(a)\cap {\mathop {\mathrm {supp}}}(c(U,{\mathop {\boldsymbol d}}(U_i))) = {\mathop {\mathrm {supp}}}(a)\cap {\mathop {\boldsymbol r}}(U_i) = {\mathop {\boldsymbol r}}(U_i)$ as $U_i =U\cdot {\mathop {\boldsymbol d}}(U_i)$ . Next observe that since $U_i\leq U$ , we have $a_i\delta _{U_i}+I =ac(U,{\mathop {\boldsymbol d}}(U_i))\delta _{U_i}+I =ac(U,{\mathop {\boldsymbol d}}(U_i))c(U,{\mathop {\boldsymbol d}}(U_i))^*\delta _U+I = a1_{{\mathop {\boldsymbol r}}(U_i)}\delta _U+I$ and so $\sum _{i=1}^k a_i\delta _{U_i}+I = \sum _{i=1}^k a1_{{\mathop {\boldsymbol r}}(U_i)}\delta _U+I = a1_{{\mathop {\boldsymbol r}}(U)}\delta _U +I = a\delta _U+I$ as the ${\mathop {\boldsymbol r}}(U_i)$ are pairwise disjoint (since U is a bisection) with union ${\mathop {\boldsymbol r}}(U)$ .

Our final lemma finds a sort of normal form for $A\rtimes _{\alpha ,c}\Gamma _c({\mathscr G})$ .

Lemma 4.13. Every element of $A\rtimes _{\alpha ,c}\Gamma _c({\mathscr G})$ is equal to one of the form $\sum _{i=1}^k a_i\delta _{U_i}+I$ , where $U_1,\ldots , U_k$ are pairwise disjoint nonempty compact open bisections and ${\mathop {\mathrm {supp}}}(a_i)= {\mathop {\boldsymbol r}}(U_i)$ , for $i=1,\ldots , k$ . Here k is possibly $0$ .

Proof. By Lemma 4.11, each element of $A\rtimes _{\alpha ,c}\Gamma _c({\mathscr G})$ can be written in the form $\sum _{j=1}^mb_j\delta _{V_j}+I$ with the $V_j\in \Gamma _c({\mathscr G})$ and ${\mathop {\mathrm {supp}}}(b_j)={\mathop {\boldsymbol r}}(V_j)$ . Since $A\delta _{\emptyset }\subseteq I$ by Proposition 4.5, we may assume that all the $V_j$ are nonempty, unless our element is $0$ , in which case there is nothing to prove. The compact open subsets of ${\mathscr G}$ form a Boolean algebra because ${\mathscr G}$ is Hausdorff. The Boolean algebra generated by $V_1,\ldots , V_m$ is finite and, hence, is generated by its atoms $W_1,\ldots , W_k$ . Moreover, since $V=V_1\cup \cdots \cup V_m$ is the maximum of this Boolean algebra, we have $W_i=V\cap W_i= (V_1\cap W_i)\cup \cdots \cup (V_m\cap W_i)$ and, hence, since $W_i$ is an atom, we must have $W_i=V_j\cap W_i$ for some j, that is, $W_i\subseteq V_j$ . Thus the $W_i$ are compact open bisections. Each $V_j$ is the disjoint union of the $W_i$ contained in it and so Lemma 4.12 lets us write $b_j\delta _{V_j}+I=\sum _{W_i\subseteq V_j} b_{ji}\delta _{W_i}+I$ with ${\mathop {\mathrm {supp}}}(b_{ji}) = {\mathop {\boldsymbol r}}(W_i)$ . Thus

$$ \begin{align*} \sum_{j=1}^mb_j\delta_{V_j}+I=\sum_{j=1}^m\sum_{W_i\subseteq V_i} b_{ji}\delta_{W_i}+I=\sum_{i=1}^k d_i\delta_{W_i}+I \end{align*} $$

with ${\mathop {\mathrm {supp}}}(d_i)\subseteq {\mathop {\boldsymbol r}}(W_i)$ . Using Lemma 4.11, we can replace $d_i\delta _{W_i}$ by $a_i\delta _{U_i}$ with $U_i\subseteq W_i$ and ${\mathop {\mathrm {supp}}}(a_i)={\mathop {\boldsymbol r}}(U_i)$ . As $A\delta _{\emptyset }\subseteq I$ by Proposition 4.5, we may remove also the empty $U_i$ . This completes the proof.

We are now ready to prove the main theorem of this section.

Theorem 4.14. Let R be a commutative ring, ${\mathscr G}$ a Hausdorff ample groupoid and $R^\times \times {\mathscr G}^{(0)}\xrightarrow {\,\,\iota \,\,}\Sigma \xrightarrow {\,\,{\varphi }\,\,} {\mathscr G}$ a twist. Let $j\colon \Gamma _c({\mathscr G})\to \Gamma _c(\Sigma )$ be an idempotent-preserving set-theoretic section and let $c\colon \Gamma _c({\mathscr G})\times \Gamma _c({\mathscr G})\to C_c({\mathscr G}^{(0)}, R^\times \cup \{0\})$ be the corresponding normalized $2$ -cocycle. Then the R-algebras $C_c({\mathscr G}^{(0)},R)\rtimes _{\alpha ,c} \Gamma _c({\mathscr G})$ and $A_R({\mathscr G};\Sigma )$ are isomorphic.

Proof. We retain the above notation. Define a mapping from $\bigoplus _{U\in \Gamma _c({\mathscr G})}A\delta _U$ (where $A=C_c({\mathscr G}^{(0)},R)$ ) to $A_R({\mathscr G};\Sigma )$ as follows. Let $\pi \colon R^\times \times {\mathscr G}^{(0)}\to R^\times $ be the projection. For $a\in A$ and $U\in \Gamma _c({\mathscr G})$ , put

$$ \begin{align*} f_{a,U}(s) =\begin{cases} a({\mathop{\boldsymbol r}}({\varphi}(s)))(\pi(\iota^{-1}(s(({\varphi}|_{j(U)})^{-1}({\varphi}(s)))^{-1})))^{-1} & \text{if } {\varphi}(s)\in U,\\ 0 & \text{otherwise.}\end{cases} \end{align*} $$

(This makes sense by Proposition 3.1.) Then $f_{a,U}(s)$ is $0$ , unless ${\varphi }(s)\in U$ , in which case it is $a({\mathop {\boldsymbol r}}({\varphi }(s)))t$ , where $t\in R^\times $ satisfies $ts\in j(U)$ . Since U is clopen, it is obvious that $f_{a,U}$ is continuous (that is, locally constant). Notice that ${\varphi }({\mathop {\mathrm {supp}}}(f_{a,U})) ={\mathop {\mathrm {supp}}}(a)U$ is compact. For anti-equivariance, we compute

$$ \begin{align*} f_{a,U}(ts) =\begin{cases} a({\mathop{\boldsymbol r}}({\varphi}(s)))(\pi(\iota^{-1}(ts(({\varphi}|_{j(U)})^{-1}({\varphi}(s)))^{-1})))^{-1} & \text{if } {\varphi}(s)\in U,\\ 0 & \text{otherwise,}\end{cases} \end{align*} $$

as ${\varphi }(ts)={\varphi }(s)$ . Thus $f_{a,U}(ts) =t^{-1} f_{a,U}(s)$ , and so $f_{a,U}\in A_R({\mathscr G};\Sigma )$ .

We claim that $\psi (a\delta _U) = f_{a,U}$ is a homomorphism from $\bigoplus _{U\in \Gamma _c({\mathscr G})}A\delta _U$ onto $A_R({\mathscr G};\Sigma )$ . It is clearly R-linear. We first compute $f_{a,U}\ast f_{b,V}(s)$ . We choose a section $p\colon {\mathscr G}\to \Sigma $ such that $p|_U = ({\varphi }|_{j(U)})^{-1}$ . Then in order for $f_{a,U}\ast f_{b,V}(s)\neq 0$ we need ${\varphi }(s)\in UV$ . Suppose this is the case and put ${\varphi }(s)=gh$ with $g\in U$ and $h\in V$ . Let $\widetilde {g}=({\varphi }|_{j(U)})^{-1}(g)$ and $\widetilde {h} =({\varphi }|_{j(V)})^{-1}(h)$ . Put $\widetilde {s} = ({\varphi }|_{j(UV)})^{-1}(s)$ . Then we can write $s=t\widetilde {s}$ with $t\in R^\times $ and $\widetilde {g}{\kern2pt}\widetilde {h} = c(U,V)({\mathop {\boldsymbol r}}({\varphi }(s)))\widetilde {s}$ . We compute that

$$ \begin{align*} f_{a,U}\ast f_{b,V}(s) = a({\mathop{\boldsymbol r}}(g))b({\mathop{\boldsymbol r}}(h))\pi(\iota^{-1}({\kern1pt}\widetilde{g}^{-1} s\widetilde{h}^{-1}))^{-1}. \end{align*} $$

But $\widetilde {g}^{-1} s\widetilde {h}^{-1} = t\widetilde {g}^{-1} \widetilde {s}\widetilde {h}^{-1} = tc(U,V)({\mathop {\boldsymbol r}}({\varphi }(s)))^{-1}{\mathop {\boldsymbol r}}(h)$ . Thus

$$ \begin{align*} f_{a,U}\ast f_{b,V}(s) =a({\mathop{\boldsymbol r}}(g))b({\mathop{\boldsymbol r}}(h))t^{-1} c(U,V)({\mathop{\boldsymbol r}}({\varphi}(s))). \end{align*} $$

On the other hand, $a\delta _Ub\delta _V = a(Ub)c(U,V)\delta _{UV}$ . Now $f_{a(Ub)c(U,V),UV}(s)$ is $0$ unless ${\varphi }(s)\in UV$ . Then, retaining the previous notation,

$$ \begin{align*} f_{a(Ub)c(U,V),UV}(s) &= a({\mathop{\boldsymbol r}}({\varphi}(s)))(Ub)({\mathop{\boldsymbol r}}({\varphi}(s)))c(U,V)({\mathop{\boldsymbol r}}({\varphi}(s)))\pi(\iota^{-1}(s\widetilde{s}^{{\kern2pt}-1}))^{-1}\\[-3pt] &= a({\mathop{\boldsymbol r}}(g))b({\mathop{\boldsymbol r}}(h))c(U,V)({\mathop{\boldsymbol r}}({\varphi}(s)))t^{-1}. \end{align*} $$

This completes the proof that $\psi $ is a homomorphism.

If $V\subseteq \Sigma $ is a compact open bisection (whence ${\varphi }|_V$ is injective by Propositions 3.1), let $\widetilde {1}_V\in A_R(G;\Sigma )$ be given by

$$ \begin{align*} \widetilde{1}_V(s) = \begin{cases}\pi(\iota^{-1}(({\varphi}|_V)^{-1}({\varphi}(s))s^{-1})) & \text{if } {\varphi}(s)\in {\varphi}(V),\\[-3pt] 0 &\text{otherwise,}\end{cases} \end{align*} $$

that is, $\widetilde {1}_V$ is supported on $R^\times \cdot V$ and sends s to t if $ts\in V$ satisfies $t\in R^\times $ . It is shown in [4, Proposition 2.8] that $A_R({\mathscr G};\Sigma )$ is spanned as an R-module by the $\tilde {1}_V$ . So, to show that $\psi $ is onto, we just need to show that $\widetilde {1}_V$ is in the image of $\psi $ for any $V\in \Gamma _c(\Sigma )$ . Let $U={\varphi }(V)\in \Gamma _c({\mathscr G})$ and note that ${\varphi }|_{j(U)}\colon j(U)\to U$ is a homeomorphism by Proposition 3.1. Then we can define $a\in A$ by

$$ \begin{align*} a(x) = \begin{cases} \pi(\iota^{-1}(({\varphi}|_V)^{-1} (({\mathop{\boldsymbol r}}|_U)^{^{-1}}(x))({\varphi}|_{j(U)})^{-1} (({\mathop{\boldsymbol r}}|_U)^{^{-1}}(x))^{-1})) & \text{if }x\in {\mathop{\boldsymbol r}}(U),\\[-3pt] 0 & \text{otherwise.}\end{cases} \end{align*} $$

Note that a is supported on ${\mathop {\boldsymbol r}}(U)$ and so belongs to A. We claim that $f_{a,U} = \widetilde {1}_V$ . Both functions vanish on all $s\in \Sigma $ with ${\varphi }(s)\notin U$ . If ${\varphi }(s)\in U$ , let $t_1,t_2\in R^\times $ satisfy $t_1s\in j(U)$ and $t_2s\in V$ . Then $\widetilde {1}_V(s)=t_2$ . On the other hand, $f_{a,U}(s) = a({\mathop {\boldsymbol r}}({\varphi }(s)))t_1=t_2t_1^{-1} t_1=t_2$ , as required. We conclude that $\psi $ is onto.

We next show that $I=\ker \psi $ . Suppose that $a\in A$ and $U\leq V\in \Gamma _c({\mathscr G})$ . We need to show that $f_{a,U} = f_{ac(V,{\mathop {\boldsymbol d}}(U))^*,V}$ . Note that ${\mathop {\mathrm {supp}}}(c(V,{\mathop {\boldsymbol d}}(U))^*) = {\mathop {\boldsymbol r}}(V\cdot {\mathop {\boldsymbol d}}(U))={\mathop {\boldsymbol r}}(U)$ . Thus both functions vanish on any $s\in \Sigma $ with ${\varphi }(s)\notin U$ . Assume that ${\varphi }(s)\in U$ . Let $\widetilde {s}\in j(U)$ satisfy ${\varphi }({\kern2pt}\widetilde {s}{\kern2pt})={\varphi }(s)$ and $\overline {s}\in j(V)$ satisfy ${\varphi }(\overline {s})={\varphi }(s)$ . Since $j({\mathop {\boldsymbol d}}(U)) = {\mathop {\boldsymbol d}}(U)$ , we have

$$ \begin{align*} j(V){\mathop{\boldsymbol d}} (U) = \{\iota(c(V,{\mathop{\boldsymbol d}}(U))({\mathop{\boldsymbol r}}(g)),{\mathop{\boldsymbol r}}(g))\mid g\in U\}j(U), \end{align*} $$

and so $\overline {s}=c(V,{\mathop {\boldsymbol d}}(U))({\mathop {\boldsymbol r}}({\varphi }(s)))\widetilde {s}$ . If $s=u\widetilde {s}$ satisfies $u\in R^\times $ , then

$$ \begin{align*} s=uc(V,{\mathop {\boldsymbol d}}(U)) ({\mathop {\boldsymbol r}}({\varphi }(s)))^{-1} \overline {s}. \end{align*} $$

Therefore, $f_{a,U}(s) =a({\mathop {\boldsymbol r}}({\varphi }(s)))u^{-1}$ and

$$ \begin{align*} f_{ac(V,{\mathop{\boldsymbol d}}(U))^*,V}(s) &= a({\mathop{\boldsymbol r}}({\varphi}(s)))(c(V,{\mathop{\boldsymbol d}}(U))({\mathop{\boldsymbol r}}({\varphi}(s))))^{-1} c(V,{\mathop{\boldsymbol d}}(U))({\mathop{\boldsymbol r}}({\varphi}(s)))u^{-1}\\[-3pt] &=a({\mathop{\boldsymbol r}}({\varphi}(s)))u^{-1}, \end{align*} $$

as required. Thus $I\subseteq \ker \psi $ .

Let $z\in \bigoplus _{U\in \Gamma _c({\mathscr G})}A\delta _U$ belong to $\ker \psi $ . Then $z+I = \sum _{i=1}^k a_i\delta _{U_i}+I$ , where ${\mathop {\mathrm {supp}}}(a_i)={\mathop {\boldsymbol r}}(U_i)$ , the $U_i$ are pairwise disjoint and nonempty by Lemma 4.13, and possibly $k=0$ . Since $I\subseteq \ker \psi $ , we conclude that $0=\psi (z)=\sum _{i=1}^k f_{a_i, U_i}$ . Since the $U_i$ are pairwise disjoint, we deduce that each $f_{a_i, U_i}=0$ . Since ${\mathop {\mathrm {supp}}}(a_i)={\mathop {\boldsymbol r}}(U_i)$ , the only way $f_{a_i, U_i}$ vanishes is if $U_i=\emptyset $ . We deduce that $k=0$ and, hence, $z\in I$ . This completes the proof.