A groupoid generalisation of Leavitt path algebras

Abstract

Let $G$ be a locally compact, Hausdorff, étale groupoid whose unit space is totally disconnected. We show that the collection $A(G)$ of locally-constant, compactly supported complex-valued functions on $G$ is a dense $*$-subalgebra of $C_c(G)$ and that it is universal for algebraic representations of the collection of compact open bisections of $G$. We also show that if $G$ is the groupoid associated to a row-finite graph or $k$-graph with no sources, then $A(G)$ is isomorphic to the associated Leavitt path algebra or Kumjian–Pask algebra. We prove versions of the Cuntz–Krieger and graded uniqueness theorems for $A(G)$.

1 Introduction

A ring $R$ is said to have invariant basis number if any two bases (i.e., $R$-linearly independent spanning sets) of a free left $R$-module have the same number of elements. Many familiar rings (e.g., fields, commutative rings, left-Noetherian rings) have invariant basis number, but there are many examples of noncommutative rings that do not. A ring $R$ without invariant basis number is said to have module type $(m,n)$ if $m < n$ are natural numbers chosen minimally with $R^m \cong R^n$ as left $R$-modules. In the 1940’s, Leavitt constructed algebras $L_{m,n}$ with module type $(m,n)$ for all pairs of natural numbers with $m < n$ [13, 14]. The $L_{m,n}$ are now known as the Leavitt algebras, and when $m = 1$, the Leavitt algebra $L_{1,n}$ is the unique nontrivial unital complex algebra generated by elements $x_1 \dots x_n$ and $y_1, \dots , y_n$ such that $\sum ^n_{i=1} x_i y_i = 1$ and $y_i x_j = \delta _{i,j} 1$ for all $i,j \le n$. In the 1970’s, independent of Leavitt’s work and motivated by the search for $C^{*}$-algebraic analogues of Type III factors, Cuntz defined a class of $C^{*}$-algebras $\mathcal O_n$, one for each integer $n \ge 2$, which are generated by elements $s_1, \dots , s_n$ satisfying $\sum ^n_{i=1} s_i s^*_i = 1$ and $s^*_i s_i = 1$ for all $i$ (it follows that $s^*_i s_j = \delta _{i,j} 1$ for all $i,j \le n$). A consequence of the uniqueness of $L_{1,n}$ is that it is isomorphic to the dense $*$-subalgebra of $\mathcal O_n$ generated by $s_1, \dots , s_n$ via an isomorphism that carries each $x_i$ to $s_i$ and each $y_i$ to $s^*_i$.

Shortly after Cuntz’s work, Cuntz and Krieger [6] generalised Cuntz’s results to describe a class of $C^*$-algebras $\mathcal O_A$ associated to binary-valued matrices $A$. At about the same time, Enomoto and Watatani provided a very elegant description of these Cuntz–Krieger algebras in terms of the directed graphs encoded by the matrices. Nearly twenty years later, Kumjian et al. [12] developed the class of $C^*$-algebras now known as graph $C^*$-algebras, as a far-reaching generalisation of the Cuntz–Krieger algebras patterned on Enomoto and Watatani’s approach. Each graph $C^*$-algebra is described in terms of generators associated to the vertices and edges in the graph subject to relations encoded by connectivity in the graph. The Cuntz algebra $\mathcal O_n$ corresponds to the graph with one vertex and $n$ edges. A remarkable assortment of important $C^*$-algebraic properties of a graph $C^*$-algebra can be characterised in terms of the structure of the graph (see [15] for a good overview). Shortly afterwards, Kumjian and Pask [10] introduced a sort of higher-dimensional graph, now known as a $k$-graph, and an associated class of $C^*$-algebras, as a flexible visual model for the higher-rank Cuntz–Krieger algebras discovered by Robertson and Steger [20]. When $k = 1$, a $k$-graph is essentially a directed graph, and Kumjian and Pask’s $C^*$-algebras coincide with the graph $C^*$-algebras of [12].

In the early 2000’s, the algebraic community became interested in the similarity between the constructions of Leavitt and Cuntz and the potential for the graph $C^*$-algebra template to provide a broad class of interesting new algebras. Following the lead of [12], Abrams and Aranda Pino associated Leavitt path algebras to a broad class of directed graphs. The Leavitt path algebra of a directed graph is the universal algebra whose presentation in terms of generators and relations is essentially the same as that of the graph $C^*$-algebra. Moreover, the graded uniqueness theorem for Leavitt path algebras implies that the $C^*$-algebra of a directed graph is a norm completion of its Leavitt path algebra [16, 23]. Further generalising Leavitt path algebras, Aranda Pino et al. [3] recently constructed a class of algebras associated to $k$-graphs, which they call Kumjian–Pask algebras.

A very powerful framework for constructing $C^*$-algebras is the notion of a groupoid $C^*$-algebra. Renault’s structure theory for groupoid $C^*$-algebras [19] is exploited in [12] where structural properties of the graph $C^{*}$-algebra are deduced by showing that the graph $C^{*}$-algebra is isomorphic to a groupoid $C^*$-algebra and then tapping into Renault’s results [19]. The same approach was taken in [10] to establish important structural properties of $k$-graph $C^*$-algebras: the $C^*$-algebra of a $k$-graph is defined in terms of generators and relations, but its structure is analysed by identifying it with a groupoid $C^*$-algebra.

In this paper, from a sufficiently well-behaved groupoid $G$, we construct a complex algebra $A(G)$ with the following properties:

• (1) $A(G)$ has a natural description as a universal algebra (Theorem 3.10);

• (2) $A(G)$ is isomorphic to a dense subalgebra of the groupoid $C^{*}$-algebra $C^*(G$) (Proposition 4.2); and

• (3) given a $k$-graph $\Lambda$, if $G= G_\Lambda$ is the groupoid corresponding to $\Lambda$ as in [10] (Proposition 4.3), then $A(G)$ is isomorphic to the Kumjian–Pask algebra $\mathrm{KP }_\mathbb {C}(\Lambda )$. In particular, if $E$ is a directed graph and $G= G_E$ is the graph groupoid associated to $E$, then $A(G)$ is isomorphic to the Leavitt path algebra $L_\mathbb {C}(E)$.

In [22], Steinberg defines a groupoid algebra $KG$ for an arbitrary commutative ring $K$ with unit and shows that $KG$ is a quotient of an associated inverse semigroup algebra. We show that the algebra $A(G)$ is identical to $KG$ for $K = \mathbb {C}$ (the complex numbers).¹ Our approach is different from that of [22] and our universal property and uniqueness theorems (see below) provide new tools for studying $KG$ and the inverse semigroup algebras associated to them in the case where $K= \mathbb {C}$; it would be interesting to investigate versions of these theorems for general $K$.

The Cuntz–Krieger uniqueness theorem and gauge-invariant uniqueness theorem are important tools in the study of graph $C^*$-algebras. Versions of these theorems have been established for many generalisations of Cuntz–Krieger algebras [5, 8, 10–12, 17, 18]. For Leavitt path algebras, the graded uniqueness theorem is the analogue of the gauge-invariant uniqueness theorem. The first version of this graded uniqueness theorem was a corollary to Ara, Moreno, and Pardo’s characterisation [2, Theorem 4.3] of the graded ideals in a Leavitt path algebra. It was first stated explicitly by Raeburn who proved both the graded uniqueness theorem and Cuntz–Krieger uniqueness theorem for Leavitt path algebras of row-finite graphs with no sinks and over fields equipped with a positive definite $*$-operation [16, Theorem 1.3.2 and Theorem 1.3.4]. Tomforde extended these results to Leavitt path algebras of arbitrary graphs over arbitrary fields in [23, Theorem 4.8 and Theorem 6.8], and later proved the two uniqueness theorems for Leavitt path algebras of arbitrary graphs over a ring [24, Theorem 5.3 and Theorem 6.5]. Aranda Pino et al. [3] subsequently proved versions of these theorems for Kumjian–Pask algebras. In Sect. 5 we prove versions of the Cuntz–Krieger uniqueness theorem (Theorem 5.1) and the graded uniqueness theorem (Theorem 5.4) for $A(G)$. We also give an example of a groupoid satisfying our hypothesis that is not necessarily the groupoid of a $k$-graph.

Our aim in defining and initiating the analysis of $A(G)$ is twofold: (1) to provide a broad framework for future generalisations of Leavitt path algebras from other combinatorial structures; and (2) to make available the powerful toolkit of groupoid analysis to study these algebras. In addition, we hope this will provide a new and useful perspective on the interplay between algebra and analysis at the interface between Leavitt path algebras and graph $C^*$-algebras.

2 Preliminaries

A groupoid is a small category with inverses. We write $G^{(2)}\subseteq G\times G$ for the set of composable pairs in $G$; we write $G^{(0)}$ for the unit space of $G$, and we denote by $r$ and $s$ the range and source maps $r,s : G\rightarrow G^{(0)}$. So $(\alpha ,\beta ) \in G^{(2)}$ if $s(\alpha ) = r(\beta )$. For $U,V \subseteq G$, we define

$$\begin{aligned} UV:= \{ \alpha \beta : \alpha \in U, \ \beta \in V, \text { and } r(\beta ) = s(\alpha ) \}. \end{aligned}$$

(2.1)

A topological groupoid is a groupoid endowed with a topology under which $r$ and $s$ are continuous, the inverse map is continuous, and such that composition is continuous with respect to the relative topology on $G^{(2)}$ inherited from $G\times G$.

Recall that if $G$ is a groupoid, then an open bisection of $G$ is an open subset $U \subseteq G$ such that $r|_U$ and $s|_U$ are homeomorphisms. We will work exclusively with locally compact, Hausdorff groupoids which are étale in the sense that the source map $s : G\rightarrow G^{(0)}$ is a local homeomorphism. The range map is then a local homeomorphism as well. If $G$ is étale then $G^{(0)}$ is open in $G$ and $G$ admits a Haar system consisting of counting measures. The following also appears as [7, Proposition 4.1].

Lemma 2.1

Let $G$ be a locally compact, Hausdorff, étale groupoid. Suppose that $G^{(0)}$ is totally disconnected. Then the topology on $G$ has a basis of clopen bisections. Moreover, if $G$ is locally compact and Hausdorff, then $G$ has a basis of compact open bisections.

Proof

Proposition 2.8 of [19] implies that $G$ has a basis of open bisections. For each $\gamma \in G$, let $U$ be an open bisection containing $\gamma$. Since $r$ is an open map there exists a basic clopen neighbourhood $X$ of $r(\gamma )$ such that $X \subseteq r(U)$. Then $XU = \{h \in U : r(h) \in X\} = U \cap r^{-1} (X)$ is homeomorphic to $X$ by choice of $U$ and in particular is a clopen bisection containing $\gamma$. If $G$ is also locally compact, then $U$ may be chosen to be precompact. Hence the clopen subset $XU$ is a compact open bisection.$\square$

Notation 2.2

For the remainder of this paper, $\Gamma$ will denote a discrete group, $G$ will denote a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c$ will denote a continuous cocycle from $G$ to $\Gamma$ (that is, $c$ carries composition in $G$ to the group operation in $\Gamma$).

By Lemma 2.1, with $\Gamma$, $G$ and $c$ as above, $G$ has a basis of compact open bisections. Since $G$ is Hausdorff, compact subsets of $G$ are closed. We will use this fact frequently and without further comment.

Remark 2.3

These hypotheses might sound very restrictive, but, for instance, every $k$-graph groupoid satisfies them (see, for example, [8]).

Remark 2.4

Let $U$ be a compact open subset of a topological space $X$. Let $F$ be a finite cover of $U$ by compact open subsets of $U$. For each nonempty $H \subseteq F$, let $V_H := \big (\bigcap H\big ) \setminus \big (\bigcup (F \setminus H)\big )$. Since each $V \in F$ is compact and open, so is each $V_H$. In particular, since $F$ is finite, so is $K := \{H \subseteq F : H \not = \emptyset , V_H \not = \emptyset \}$, and

$$\begin{aligned} U = \bigsqcup _{H \in K} V_H \end{aligned}$$

is an expression for $U$ as a finite disjoint union of nonempty compact open sets such that for each $W \in K$ we have $W \subseteq V$ for at least one $V \in F$, and such that whenever $W \in K$ and $V \in F$ satisfy $W \not \subseteq V$, we have $W \cap V = \emptyset$. We refer to this as the disjointification of the cover $F$ of $U$.

Throughout this paper, unless stated otherwise, all algebras are taken to be complex $^*$-algebras, and all representations are assumed to preserve adjoints.

3 The algebra $A(G)$

Definition 3.1

Let $X$ be a topological space. A function $f : X \rightarrow Y$ is locally constant if for every $x \in X$ there exists a neighbourhood $U$ of $x$ such that $f|_U$ is constant.

Observe that if $f : X \rightarrow \mathbb {C}$ is locally constant then it is automatically continuous, and the support of $f$ is clopen in $X$.

Definition 3.2

Let $G$ be a locally compact, Hausdorff, étale groupoid with totally disconnected unit space. We define $A(G)$ to be the compex vector space

$$\begin{aligned} A(G)= \{f \in C_c(G) : f \text { is locally constant}\} \end{aligned}$$

with pointwise addition and scalar multiplication.

The following lemma shows that $A(G)$ is precisely the algebra $\mathbb {C}G$ of [22, Definition 4.1]. (In fact, Definition 3.2 agrees precisely with the definition of $\mathbb {C}G$ given in the preprint version of [22]—see [21, Definition 3.1]—and then the following Lemma is [21, Proposition 3.3]).

Lemma 3.3

Let $\Gamma$ be a discrete group and $G$ be a locally compact, Hausdorff, étale groupoid with totally disconnected unit space. If $\mathcal U$ is the basis of all compact open subsets of $G$, we have $A(G)= \mathrm{span }\{1_U : U \in \mathcal U\}$.

Proof

For any $U\in \mathcal U$, the function $1_U$ is locally constant, and hence $\mathrm{span }\{1_U : U \in \mathcal U\} \subseteq A(G)$. We must show that $A(G)\subseteq \mathrm{span }\{1_U : U \in \mathcal U\}$. Fix $f \in A(G)$. Since $f$ is locally constant and $\mathcal U$ is a basis, for each $\alpha \in \mathrm{supp }(f)$, there is a neighbourhood $U_\alpha \in \mathcal U$ of $\alpha$ such that $f|_{U_{\alpha }}$ is constant. Since $\mathrm{supp }(f)$ is clopen, we may assume that $U_\alpha \subseteq \mathrm{supp }(f)$. Since $\mathrm{supp }(f)$ is compact there is a finite subset $F \subseteq \{U_{\alpha }\}_{\alpha \in \mathrm{supp }f}$ such that $\mathrm{supp }(f) = \bigcup F$. Let $K$ be the disjointification of $F$ discussed in Remark 2.4. Since $f$ is constant on each $V \in F$ and each $W \in K$ is a subset of some $V \in F$, the function $f$ is constant on each $W \in K$. Hence, writing $f(W)$ for the unique value taken by $f$ on $W \in K$, we have $f = \sum _{W \in K} f(W) 1_{W}$.$\square$

Definition 3.4

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. For $n \in \Gamma$ we write $G_n := c^{-1}(n)$. We write $A_n(G)$ for the subset of $A(G)$ consisting of functions whose support is contained in $G_n$. We say that a subset $S$ of $G$ is graded if the cocycle $c$ is constant on $S$. If $S \subseteq G_n$, we say that $S$ is $n$ -graded. For each $n \in \Gamma$ we write $B^{\mathrm{co }}_{n}(G)$ for the collection of all $n$-graded compact open bisections of $G$. We write $B^{\mathrm{co }}_{*}(G)$ for $\bigcup _{n \in \Gamma } B^{\mathrm{co }}_{n}(G)$.

Lemma 3.5

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. We have $A(G)= \mathrm{span }\{A_n : n \in \Gamma \} \subseteq C_c(G)$. Every $f \in A(G)$ can be expressed as $f = \sum _{U \in F} a_U 1_{U}$ where $F$ is a finite subset of $B^{\mathrm{co }}_{*}(G)$ whose elements are mutually disjoint and $a : U \mapsto a_U$ is a function from $F$ to $\mathbb {C}$.

Proof

We have $A(G)\supseteq \mathrm{span }\{A_n : n \in \Gamma \}$ because each $A_n$ consists of locally constant functions. For the reverse inclusion, fix $f \in A(G)$. Since $\Gamma$ is discrete and $c$ is continuous, each $G_n$ is clopen. Since $\mathrm{supp }(f)$ is compact, there is a finite collection $N \subseteq G$ such that $\mathrm{supp }(f) \subseteq \bigcup _{n \in N} G_n$. For $n \in N$ let $f_n$ denote the pointwise product $f 1_{G_n}$. Then $f_n$ is locally constant and continuous because $1_{G_n}$ and $f$ are. We then have $f = \sum _{n \in N} f_n \in \mathrm{span }\{A_n : n \in \Gamma \}$.

Let $f \in A(G)$. By Lemma 3.3, there is a finite set $K_0$ of compact open sets and an assignment $W \mapsto d_W$ of scalars to the elements of $K_0$ such that $f = \sum _{W \in K_0} d_W 1_W$. Let

$$\begin{aligned} K := \{W \cap G_n : W \in K_0, n \in \Gamma , W \cap G_n \not = \emptyset \}. \end{aligned}$$

Since $\Gamma$ is discrete and $c$ is continuous, each $G_n$ is open. Since each $W \in K_0$ is compact, $K$ is finite. Each $V \in K$ is graded; we write $c(V)$ for the unique value taken by $c$ on $V$. For each $V \in K$, let

$$\begin{aligned} b_V = \sum _{W \in K_0, W \cap G_{c(V)} = V} d_W \end{aligned}$$

Then $f = \sum _{V \in K} b_V 1_V$.

Let $F$ be the disjointification of $K$. Each $U \in F$ is graded because $F$ is a refinement of $K$. For $U \in F$, define

$$\begin{aligned} a_U = \sum _{V \in K,\, U \subseteq V} b_V. \end{aligned}$$

Then $f = \sum _{U \in F} a_U 1_U$ is the desired expression.$\square$

Recall that given a locally compact, Hausdorff, étale groupoid $G$ such that $s : G\rightarrow G^{(0)}$ is a local homeomorphism, and given $f,g \in A(G) \subseteq C_c(G)$, the functions $f^*$ and $f * g$ are given by

$$\begin{aligned} f^*(\gamma )&= \overline{f(\gamma ^{-1})} \end{aligned}$$

(3.1)

$$\begin{aligned} (f*g)(\gamma )&= \sum _{r(\alpha ) = r(\gamma )} f(\alpha ) g(\alpha ^{-1}\gamma ). \end{aligned}$$

(3.2)

Proposition 3.6

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. Under the operations (3.1) and (3.2), $A(G)$ is a $\Gamma$-graded $*$-algebra with graded subspaces $A_n$ as described in Definition 3.4.

Remark 3.7

For us, an involution on a $*$-algebra over $\mathbb {C}$ is always conjugate linear.

Remark 3.8

We do not assume that $\Gamma$ is abelian so we will write the group operation multiplicatively.

Proof

That $A(G)$ is a complex algebra follows from [22, Proposition 4.6]. We must verify that $A(G)$ is a $*$-algebra and that $A(G)$ is graded. The $A_n$ are mutually linearly independent because the $G_n$ are disjoint and restriction of functions gives a vector-space isomorphism of each $A_n$ onto the space of locally constant functions on $G_n$. Observe that the $*$-operation is a conjugate-linear involution on $A(G)$ and takes $A_n$ to $A_{n^{-1}}$. Next we will show that the multiplication defined on $A(G)$ is a graded multiplication. If $f \in A_m$ and $g \in A_n$, then if $(f * g)(\gamma ) \not = 0$ we have $f(\alpha ) \not = 0$ and $g(\alpha ^{-1}\gamma ) \not = 0$ for some $\alpha$ with $r(\alpha ) = r(\gamma )$. In particular, $c(\alpha ) = m$, and $c(\alpha ^{-1}\gamma ) = n$ forcing $c(\gamma ) = mn$ (because $c(\gamma )=c(\alpha \alpha ^{-1}\gamma )=c(\alpha )c(\alpha ^{-1}\gamma )$). Hence $\mathrm{supp }(f*g)\subseteq G_{mn}$.$\square$

We finish this section by presenting of $A(G)$ as a universal algebra.

Definition 3.9

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. Let $B$ be an algebra over $\mathbb {C}$. A representation of $B^{\mathrm{co }}_{*}(G)$ in $B$ is a family $\{t_U : U \in B^{\mathrm{co }}_{*}(G)\} \subseteq B$ satisfying

• (R1) $t_\emptyset = 0$;

• (R2) $t_Ut_V = t_{UV}$ for all $U,V \in B^{\mathrm{co }}_{*}(G)$; and

• (R3) $t_U + t_V = t_{U \cup V}$ whenever $U$ and $V$ are disjoint elements of $B^{\mathrm{co }}_{n}(G)$ for some $n$ such that $U \cup V$ is a bisection.

The following theorem gives an alternative formulation of [21, Theorem 3.11].

Theorem 3.10

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. Then $\{1_U : U \in B^{\mathrm{co }}_{*}(G)\} \subseteq A(G)$ is a representation of $B^{\mathrm{co }}_{*}(G)$ which spans $A(G)$. Moreover, $A(G)$ is universal for representations of $B^{\mathrm{co }}_{*}(G)$ in the sense that for every representation $\{t_U : U \in B^{\mathrm{co }}_{*}(G)\}$ of $B^{\mathrm{co }}_{*}(G)$ in an algebra $B$, there is a unique homomorphism $\pi : A(G)\rightarrow B$ such that $\pi (1_U) = t_U$ for all $U \in B^{\mathrm{co }}_{*}(G)$.

Proof

The collection $\{1_U : U \in B^{\mathrm{co }}_{*}(G)\}$ certainly satisfies (R1) and (R3), and it satisfies (R2) by [22, Proposition 4.5 (3)]. That this family spans $A(G)$ follows from Lemma 3.5.

Let $B$ be a complex algebra and let $\{t_U : U \in B^{\mathrm{co }}_{*}(G)\}$ be a representation of $B^{\mathrm{co }}_{*}(G)$ in $B$. We must show that there is a homomorphism $\pi : A(G)\rightarrow B$ satisfying $\pi (1_U) = t_U$ for all $U \in B^{\mathrm{co }}_{*}(G)$; uniqueness follows from the previous paragraph. We begin by showing that

$$\begin{aligned}&\sum _{U \in F} t_U = t_{\bigcup F}\hbox { for }n \in \Gamma \hbox { and finite }F \subseteq B^{\mathrm{co }}_{n}(G)\hbox { consisting of mutually disjoint}\nonumber \\&\quad \hbox {bisections such that }\bigcup F \in B^{\mathrm{co }}_{n}(G). \end{aligned}$$

(3.3)

Let $F\subseteq B^{\mathrm{co }}_{n}(G)$ be a finite collection of mutually disjoint bisections such that $\bigcup F$ is a bisection. We claim that $r(U) \cap r(V) = \emptyset$ for distinct $U,V \in F$. To see this, fix $x \in r(U)$. There exists $\alpha \in U$ such that $r(\alpha ) = x$, and this $\alpha$ is the unique element of $\bigcup F$ whose range is $x$ because $\bigcup F$ is a bisection. Since $U \cap V = \emptyset$, we have $\alpha \not \in V$ and hence $x \not \in r(V)$. So the sets $r(U)$ where $U \in F$ are mutually disjoint as claimed. Thus each $U \in F$ satisfies $U = r(U) (\bigcup F)$. A standard induction extends (R3) to finite collections of mutually disjoint compact open subsets of $G^{(0)}$. Combining this with (R2), we obtain

$$\begin{aligned} t_{\bigcup F} = t_{r(\bigcup F)} t_{\bigcup F} = \sum _{U \in F} t_{r(U)} t_{\bigcup F} = \sum _{U \in F} t_{r(U) \big (\bigcup F\big )} = \sum _{U \in F} t_U. \end{aligned}$$

We show next that the formula $\sum _{U \in F} a_U 1_U \mapsto \sum _{U \in F} a_U t_U$ is well-defined on linear combinations of indicator functions where $F \subseteq B^{\mathrm{co }}_{*}(G)$ is a finite collection of mutually disjoint bisections. It will follow from Lemma 3.5 that there is a unique linear map $\pi : A(G) \rightarrow B$ such that $\pi (1_U) = t_U$ for each $U \in B^{\mathrm{co }}_{*}(G)$. Fix $f \in A(G)$ and suppose that

$$\begin{aligned} \sum \limits _{U \in F} a_U 1_U = f = \sum \limits _{V \in H} b_V 1_V \end{aligned}$$

where each of $F$ and $H$ is a finite set of mutually disjoint elements of $B^{\mathrm{co }}_{*}(G)$. We must show that

$$\begin{aligned} \sum \limits _{U \in F} a_U t_U = \sum \limits _{V \in H} b_V t_V. \end{aligned}$$

Since the $G_n$ are mutually disjoint, for each $n \in \Gamma$ we have

$$\begin{aligned} \sum _{U \in F \cap B^{\mathrm{co }}_{n}(G)} a_U 1_U = f|_{G_n} = \sum _{V \in H \cap B^{\mathrm{co }}_{n}(G)} b_V 1_V, \end{aligned}$$

so we may assume that $F, G \subseteq B^{\mathrm{co }}_{n}(G)$ for some $n \in \Gamma$.

Let $K = \{U \cap V : U \in F, V \in H, U \cap V \not = \emptyset \}$. Then each $W \in K$ belongs to $B^{\mathrm{co }}_{n}(G)$. Moreover, for $U \in F$ we have $U = \bigsqcup \{W \in K: W \subseteq U\}$. Hence (3.3) gives $t_U = \sum _{W \in K, W \subseteq U} t_W$ for each $U \in F$; a similar decomposition holds for $t_V$ for each $V \in H$. Therefore

$$\begin{aligned} \sum _{U \in F} a_U t_U = \sum _{U \in F}\ \sum _{W \in K,\,W \subseteq U} a_U t_W = \sum _{W \in K} \left( \sum _{U \in F,\, W \subseteq U} a_U\right) t_W, \end{aligned}$$

and similarly

$$\begin{aligned} \sum _{V \in F}b_V t_V = \sum _{W \in K} \left( \sum _{V \in F,\, W \subseteq V} b_V\right) t_W. \end{aligned}$$

Fix $W \in K$. It suffices now to show that $\sum _{U \in F,\, W \subseteq U} a_U = \sum _{V \in F,\, W \subseteq V} b_V$. By definition of $K$, the set $W$ is nonempty, so let $\alpha \in W$. Then for $U \in F$, we have $\alpha \in U \implies W \cap U \not = \emptyset \implies W \subseteq U$. Since $\alpha \in W$, this implies that $\alpha \in U \iff W \subseteq U$. Hence

$$\begin{aligned} f(\alpha ) = \sum _{U \in F} a_U 1_U(\alpha ) = \sum _{U \in F,\, \alpha \in U} a_U = \sum _{U \in F,\, W \subseteq U} a_U. \end{aligned}$$

a similar calculation shows that $\sum _{V \in F,\, W \subseteq V} b_V = f(\alpha )$ as well. So there is a linear map $\pi : A(G) \rightarrow B$ such that $\pi (1_U) = t_U$ for all $U \in B^{\mathrm{co }}_{*}(G)$.

We must check that $\pi$ is a homomorphism. To see that $\pi$ is multiplicative, fix $f, g \in A(G)$. Express $f = \sum _{U \in F} a_U 1_U$ and $g=\sum _{V \in H} b_V 1_V$ where $F$ and $H$ are finite subsets of $B^{\mathrm{co }}_{*}(G)$, and calculate:

$$\begin{aligned} \pi (fg)&= \pi \left( \Big (\sum _{U \in F} a_U 1_U\Big )\Big (\sum _{V \in H} b_V 1_V\Big )\right) \\&= \pi \left( \sum _{U \in F}\sum _{V \in H} a_U b_V 1_{U} 1_{V}\right) . \end{aligned}$$

Since [22, Proposition 4.5 (3)] gives $1_{U} 1_{V} = 1_{UV}$ for all $U,V$, we then have

$$\begin{aligned} \pi (fg) = \pi \left( \sum _{U \in F}\sum _{V \in H} a_U b_V 1_{UV}\right) = \sum _{U \in F}\sum _{V \in H} a_U b_V t_{UV}. \end{aligned}$$

Each $t_{UV} = t_{U} t_{V}$ by (R2), so

$$\begin{aligned} \pi (fg) = \sum _{U \in F}\sum _{V \in H} a_U b_V t_{U}t_{V} = \left( \sum _{U \in F} a_U t_U\right) \left( \sum _{V \in H} b_V t_V\right) = \pi (f)\pi (g) \end{aligned}$$

as required.$\square$

4 $A(G)$ is dense in $C^{*}(G)$

Since our aim is to produce algebras associated to totally disconnected, locally compact, Hausdorff groupoids whose relationship to the groupoid $C^{*}$-algebra is analogous to that of Leavitt path algebras to graph $C^{*}$-algebras, we show in this section that the subalgebra $A(G)$ of $C_c(G)$ is dense in the full (and hence also the reduced) $C^*$-algebra of $G$. We could prove this as in [12, Proposition 4.1] or [22, Proposition 6.7] by using the Stone-Weierstrass theorem, but a direct argument takes about the same amount of effort.

We first prove a technical lemma.

Lemma 4.1

Let $G$ be a locally compact, Hausdorff, étale groupoid with totally disconnected unit space. Fix a compact open bisection $U$ of $G$ and suppose that $f \in C_c(G)$ is supported on $U$. Fix $\varepsilon > 0$. There exists a finite set $\mathcal {V}$ of nonempty compact open bisections of $G$ such that $U = \bigsqcup \mathcal {V}$ and such that for each $V \in \mathcal {V}$, we have $|f(\alpha ) - f(\beta )| \le \varepsilon$ for all $\alpha , \beta \in V$.

Proof

For each $\gamma \in U$ let $U_\gamma$ be a compact open neighbourhood of $\gamma$ such that $U_\gamma \subseteq U$ and $|f(\alpha ) - f(\gamma )| < \varepsilon /2$ for all $\alpha \in U_\gamma$. Since $U$ is compact, there is a finite subset $F$ of $U$ such that $\{U_\gamma : \gamma \in F\}$ covers $U$. Let $\mathcal {V}$ be the disjointification of the $U_\gamma$ as in Remark 2.4. Fix $V\in \mathcal {V}$. Then there exists $\gamma \in F$ such that $V \subseteq U_\gamma$, and then for $\alpha , \beta \in V$, we have $|f(\alpha ) - f(\beta )| \le |f(\alpha ) - f(\gamma )| + |f(\gamma ) - f(\beta )| < \varepsilon$.$\square$

To state the next proposition, we recall from [19] that for a locally compact, Hausdorff, étale groupoid $G$, the $I$-norm on $C_c(G)$ is defined as follows. For $f \in C_c(G)$, let

$$\begin{aligned} \Vert f\Vert _{I,r}:= \sup _{u \in G^{(0)}} \left\{ \sum _{r(\alpha )=u}|f(\alpha )|\right\} \quad \text {and} \quad \Vert f\Vert _{I,s}:= \sup _{u \in G^{(0)}}\left\{ \sum _{s(\alpha )=u}|f(\alpha )|\right\} . \end{aligned}$$

Then the $I$-norm of $f$ is $\Vert f\Vert _I:=\max \{\Vert f\Vert _{I,r}, \Vert f\Vert _{I,s}\}$. The $I$-norm dominates each of the universal norm, the reduced norm, and the uniform norm on $C_c(G)$. (See [19] for further details.)

Proposition 4.2

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. With notation as above, $A(G)$ is dense in $C_c(G)$ under each of the reduced norm, the universal norm, the $I$-norm, and the uniform norm.

Proof

Since the $I$-norm dominates the other three norms, it suffices to prove the result for the $I$-norm. Fix $f \in C_c(G)$ and $\varepsilon > 0$. Since $f$ has compact support, $\mathrm{supp }(f)$ can be written as a finite union of elements of $B^{\mathrm{co }}_{*}(G)$. So we can write $f =\sum _{i=1}^n f_i$ where each $f_i$ is supported on an element of $B^{\mathrm{co }}_{*}(G)$. For each $i$, apply Lemma 4.1 to $\mathrm{supp }(f_i)$ to obtain a cover $\mathcal U_i$ of the support of $f_i$ by disjoint compact open bisections such that for $U \in \mathcal U_i$, we have $|f_i(\alpha ) - f_i(\beta )| \le \varepsilon /n$ for all $\alpha , \beta \in U_i$. For each $i \le n$ and each $U \in \mathcal U_i$, fix $z_{i,U} \in f(U)$, so $|f_i(\alpha ) - z_{i,U}| \le \varepsilon /n$ for all $\alpha \in U$. Then let $g_i := \sum _{U \in \mathcal U_i} z_{i,U} 1_U$ for all $i\le n$ and define $g := \sum _{i=1}^n g_i \in A(G)$. We have

$$\begin{aligned} \Vert f - g\Vert _I \le \sum _{i=1}^n \Vert f_i - g_i\Vert _I. \end{aligned}$$

Fix $i \le n$. It suffices to show that $\Vert f_i - g_i\Vert _I \le \varepsilon /n$. Fix $u \in G^{(0)}$. Since $f_i$ is supported on a bisection, there is at most one $\alpha \in s^{-1}(u) \cap \mathrm{supp }(f_i)$. If there is no such $\alpha$, then $\sum _{s(\alpha )=u} |(f_i - g_i)(\alpha )| = 0$ and we are done. So suppose that $\alpha \in s^{-1}(u) \cap \mathrm{supp }(f_i)$. Then there is a unique $U_0 \in \mathcal U_i$ such that $\alpha \in U_0$. Therefore $\sum _{s(\alpha )=u} |(f_i - g_i)(\alpha )| = |f_i(\alpha ) - z_{i,U_0}| \le \varepsilon /n$. Since $u \in G^{(0)}$ was arbitrary, we conclude that $\Vert f_i - g_i\Vert _{I, s} \le \varepsilon /n$. A symmetric argument gives $\Vert (f_i - g_i)(\alpha )\Vert _{I, r} \le \varepsilon /n$. Hence $\Vert f_i - g_i\Vert _I \le \varepsilon / n$ as required.$\square$

Proposition 4.3

Suppose that $\Lambda$ is a row-finite, $k$-graph with no sources and that $G_\Lambda$ is the corresponding $k$-graph groupoid. Then $A(G_\Lambda )$ as constructed above is isomorphic to the Kumjian–Pask algebra $\mathrm{KP }(\Lambda ,\mathbb {C})$.

Proof

By [10, Corollary 3.5], $t_\lambda := 1_{Z(\lambda , s(\lambda ))}$ determines a Cuntz–Krieger $\Lambda$-family in $C^*(G)$. In particular, there is a Kumjian–Pask family ([3, Definition 3.1]) for $\Lambda$ determined by $t_\lambda = 1_{Z(\lambda , s(\lambda ))}$ and $t_{\lambda ^*} = 1_{Z(s(\lambda ),\lambda )}$ for all $\lambda \in \Lambda$. It follows from the universal property of $\mathrm{KP }(\Lambda ,\mathbb {C})$ that there is a homomorphism $\phi : \mathrm{KP }(\Lambda ,\mathbb {C}) \rightarrow A(G_\Lambda )$ which carries each $s_\lambda$ to $t_\lambda$ and each $s_{\lambda ^*}$ to $t_{\lambda ^*}$.

By [3, Theorem 3.4] the algebra $\mathrm{KP }(\Lambda , \mathbb {C})$ is spanned by the elements $t_\mu t_{\nu ^*}$ where $\mu ,\nu \in \Lambda$ with $s(\mu ) = s(\nu )$, and the $\mathbb {Z}$-grading of $\mathrm{KP }(\Lambda ,\mathbb {C})$ carries each $s_\mu s_{\nu ^*}$ to $d(\mu )-d(\nu )$. So to see that $\phi$ is graded, it suffices to show that it preserves the grading of each $s_\mu s_{\nu ^*}$, which it does since

$$\begin{aligned} \phi (s_\mu s_{\nu ^*}) = 1_{Z(\mu , \nu )} = 1_{\{(\mu x, d(\mu )-d(\nu ), \nu x) : x \in \Lambda ^\infty , r(x)=s(\mu )}\} \in A_{d(\mu )-d(\nu )}. \end{aligned}$$

Since each $Z(v)$ is nonempty, $\phi (p_v) \ne 0$ for each $v \in E^0$. Thus the graded uniqueness theorem for Kumjian–Pask algebras [3, Theorem 4.1] implies that $\phi$ is injective.

It remains to show that $\phi$ is surjective. By Lemma 3.3, $A(G_\Lambda )$ is spanned by the functions $1_U$ where $U$ ranges over all compact open bisections in $G_\Lambda$. Let $U$ be a compact open bisection. Since the grading is continuous and $U$ is compact, we can write $1_U$ as the finite sum $\sum _{U \cap G_n \not = \emptyset } 1_{U \cap G_n}$ where each $U \cap G_n$ is a graded compact open bisection. So fix $n \in \mathbb {N}^k$ and a compact open $n$-graded bisection $V$. It suffices to show that $1_V \in \mathrm{span }\{1_{Z(\mu ,\nu )} : s(\mu ) = s(\nu )\}$. Because $V$ is compact and the sets $Z(\mu ,\nu )$ form a basis for the topology on $G_{\Lambda }$ [10, Proposition 2.8], we can write $V = \bigcup _{(\mu ,\nu ) \in F} Z(\mu ,\nu )$ for some finite set $F \subseteq \{(\mu , \nu ) \in \Lambda \times \Lambda : s(\mu ) = s(\nu )\}$. Since $V$ is $n$-graded, we have $d(\mu ) - d(\nu ) = n$ for all $(\mu ,\nu ) \in F$. Let $p := \bigvee _{(\mu ,\nu ) \in F} d(\mu )$. Then for each $(\mu ,\nu ) \in F$ we have $Z(\mu ,\nu ) = \bigcup \{Z(\mu \alpha , \nu \alpha ) : \alpha \in s(\mu )\Lambda ^{p - d(\mu )}\}$. Let ${H} := \{(\mu \alpha ,\nu \alpha ) : (\mu ,\nu ) \in F, \alpha \in s(\mu )\Lambda ^{p - d(\mu )}\}$. Then $Z(\eta ,\zeta ) \cap Z(\eta ',\zeta ') = \emptyset$ for distinct $(\eta ,\zeta ), (\eta ',\zeta ') \in {H}$, so $V = \bigsqcup _{(\eta ,\zeta ) \in {H}} Z(\eta ,\zeta )$. Hence $1_U = \sum _{(\eta , \zeta ) \in {H}} 1_{Z(\eta , \zeta )}$, and it follows that $\phi$ is surjective.$\square$

Remark 4.4

When $k = 1$ in the preceding proposition, $\Lambda$ is the path category of the directed graph $E = (\Lambda ^0, \Lambda ^1, r, s)$ and, in this case, the proposition specialises to the statement that $A(G)$ is isomorphic to the Leavitt path algebra of [1].

5 The uniqueness theorems

Interestingly, in the situation of groupoids, the graded uniqueness theorem is a corollary of the natural generalisation of the Cuntz–Krieger uniqueness theorem. This in turn is essentially Renault’s structure theorem for the reduced $C^*$-algebra of a groupoid in which the units with trivial isotropy are dense in the unit space. This condition has been referred to, variously, as “topologically free”, “topologically principal”, “essentially free.”

Given a unit $u$, it is standard to denote the isotropy subgroup $\{\alpha \in G : r(\alpha ) = s(\alpha ) = u\}$ by either $G(u)$ or $G^u_u$. Here we have chosen the more suggestive notation $uGu$, which is in keeping with the notation established in (2.1). Likewise, we write $Gu$ for $s^{-1}(u)$.

Theorem 5.1

Let $G$ be a locally compact, Hausdorff, étale groupoid with totally disconnected unit space. Suppose that $\{u \in G^{(0)}: uGu = \{u\}\}$ is dense in $G^{(0)}$. Let $\pi : A(G)\rightarrow B$ be a $^*$-homomorphism into a complex $^*$-algebra $B$. Suppose that $\mathrm{ker }(\pi ) \not = \{0\}$. Then there is a compact open subset $K \subseteq G^{(0)}$ such that $\pi (1_K) = 0$.

Remark 5.2

To see why the hypothesis that the units with trivial isotropy are dense is needed in Theorem 5.1, consider the situation where $G= \mathbb {Z}/2\mathbb {Z}$ regarded as a groupoid with one unit $0$. Then $A(G)$ is the group algebra $\mathbb {C}\delta _0 + \mathbb {C}\delta _1$, and the map $\pi : A(G)\rightarrow \mathbb {C}$ such that $\pi (\delta _0) = \pi (\delta _1) = 1$ is a $^*$-homomorphism of $A(G)$ which is not injective, but which restricts to an injective representation of $C_c(G^{(0)}) = \mathbb {C}\delta _0$. A related construction applies for arbitrary $G$—see [4, Proposition 4.4].

To prove Theorem 5.1, we need a technical lemma.

Lemma 5.3

Let $G$ be a locally compact, Hausdorff, étale groupoid. Fix $\alpha \in G$ and a precompact neighbourhood $V$ of $\alpha$. Suppose that $r(\alpha ) Gs(\alpha ) = \{\alpha \}$. Then there exist neighbourhoods $X$ of $r(\alpha )$ and $Y$ of $s(\alpha )$ such that ${ XVY}$ is a precompact open bisection.

Proof

Suppose, to the contrary, that for every neighbourhood $X$ of $r(\alpha )$ and every neighbourhood $Y$ of $s(\alpha )$, ${ XVY}$ fails to be a bisection. Let $U$ be an open bisection containing $\alpha$. Fix a fundamental sequence of neighbourhoods $(Y_i)^\infty _{i=1}$ of $s(\alpha )$, and for each $i$, let $X_i := r(U Y_i)$, so that $(X_i )_{i=1}^\infty$ forms a fundamental sequence of neighbourhoods of $r(\alpha )$. Since each $X_i V Y_i$ fails to be a bisection, for each $i$ there exist $\beta _i, \gamma _i \in X_i V Y_i$ with $\beta _i \ne \gamma _i$ such that either $s(\beta _i) = s(\gamma _i)$ or $r(\beta _i) = r(\gamma _i)$ for all $i$. The sequence $\big ((\beta _i, \gamma _i)\big )^\infty _{i=1}$ belongs to the precompact set $V \times V$, so by passing to a subsequence and relabelling we may assume that $\beta _i \rightarrow \beta$ and $\gamma _i \rightarrow \gamma$. Since the $X_i$ and $Y_i$ are fundamental sequences of neighbourhoods, it follows that $r(\beta _i), r(\gamma _i) \rightarrow r(\alpha )$ and $s(\beta _i), s(\gamma _i) \rightarrow s(\alpha )$. Since $r,s : G\rightarrow G^{(0)}$ are continuous and $G^{(0)}$ is Hausdorff, $r(\beta ) = r(\alpha ) = r(\gamma )$ and $s(\beta ) = s(\alpha ) = s(\gamma )$. By hypothesis, $s(\alpha ) Gr(\alpha ) = \{\alpha \}$, so we have $\beta = \gamma = \alpha$. Since $U$ is a neighbourhood of $\alpha$, we then have $\beta _i, \gamma _i \in U$ for large $i$. Fix $i$ such that $\beta _i, \gamma _i \in U$. Then $\beta _i \not = \gamma _i$ but either $r(\beta _i) = r(\gamma _i)$ or $s(\beta _i) = s(\gamma _i)$, contradicting that $U$ is a bisection.$\square$

Proof of Theorem 5.1

Fix $f \in \mathrm{ker }(\pi ) \setminus \{0\}$. Since $s$ is a local homeomorphism, it is an open map, and since $f$ is locally constant, we deduce that $s(\mathrm{supp }(f)) \subseteq G^{(0)}$ open. Because $\{u \in G^{(0)}: uGu = \{u\}\}$ is dense in $G^{(0)}$, there exists $u \in s(\mathrm{supp }(f))$ such that $uGu = \{u\}$. Fix $\alpha \in \mathrm{supp }(f)$ with $s(\alpha ) = u$. Then $r(\alpha )Gs(\alpha ) = \alpha (\alpha ^{-1} Gu) \subseteq \alpha (uGu) = \{\alpha \}$.

By Lemma 5.3, there exist compact open neighbourhoods $X$ of $r(\alpha )$ and $Y$ of $s(\alpha )$ such that $X\mathrm{supp }(f)Y$ is a bisection containing $\alpha$. Because $r$ and $s$ are continuous, $X \mathrm{supp }(f) Y= r^{-1}(X) \cap \mathrm{supp }(f) \cap s^{-1}(Y)$ is compact. Since $f$ is locally constant, $X \mathrm{supp }(f) Y$ is also open and there exist subneighbourhoods $X_0 \subseteq r(X \mathrm{supp }(f) Y)$ of $r(\alpha )$ and $Y_0 \subseteq s(X \mathrm{supp }(f) Y)$ of $s(\alpha )$ such that $X_0 \mathrm{supp }(f) Y_0$ is a compact open bisection and $f(\beta ) = f(\alpha )$ for all $\beta \in X_0 \mathrm{supp }(f) Y_0$.

We have $1_{X_0}, 1_{Y_0} \in A(G)$. By Lemma 3.3, $f$ may be written as a linear combination of characteristic functions of compact open bisections. [22, Proposition 4.5 (3)] together with bilinearity of multiplication implies that for $\beta \in G$,

$$\begin{aligned} (1_{X_0} * f * 1_{Y_0})(\beta )&= 1_{X_0}(r(\beta )) f(\beta ) 1_{X_0}(s(\beta ))\\&= 1_{X_0 \mathrm{supp }(f) Y_0}(\beta ) f(\beta ) = 1_{X_0 \mathrm{supp }(f) Y_0}(\beta ) f(\alpha ). \end{aligned}$$

Thus $f_0 := 1_{X_0} * f * 1_{Y_0} = f(\alpha ) 1_{X_0 \mathrm{supp }(f) Y_0}$. Since $\pi (f)=0$, we have $\pi (f_0) = 0$. We have $(X_0 \mathrm{supp }(f) Y_0)^{-1}(X_0 \mathrm{supp }(f) Y_0) = Y_0$ because $X_0 \mathrm{supp }(f) Y_0$ is a bisection. Proposition 4.5 (3) [22] implies that

$$\begin{aligned} f^*_0 * f_0 = |f(\alpha )|^2 1_{(X_0 \mathrm{supp }(f) Y_0)^{-1}(X_0 \mathrm{supp }(f) Y_0)} = |f(\alpha )|^2 1_{Y_0}. \end{aligned}$$

Hence $K := Y_0$ satisfies $\pi (1_K) = \frac{1}{|f(\alpha )|^2} \pi (f^*_0 * f_0) = 0$ as required.$\square$

Our graded uniqueness theorem now follows from a bootstrapping argument.

Theorem 5.4

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. Suppose that $\{u \in G^{(0)}: uG_e u = \{u\}\}$ is dense in $G^{(0)}$. Let $B$ be a complex $^*$-algebra and let $\pi : A(G)\rightarrow B$ be a graded $^*$-homomorphism. Suppose that $\mathrm{ker }(\pi ) \not = \{0\}$. Then there is a compact open subset $K \subseteq G^{(0)}$ such that $\pi (1_K) = 0$.

Proof

We first claim that there exists nonzero $f \in A_e$ such that $\pi (f) = 0$. To see this, choose $g \in \mathrm{ker }(\pi ) \setminus \{0\}$. Since $g$ is an element of the graded algebra $A(G)$, $g$ can be expressed as a finite sum of graded components $g = \sum _{h \in F} g_h$ where $F \subseteq \Gamma$ and each $g_h \in A_h$. Now $\pi (g) = \sum _{h \in F} \pi (g_h) = 0$, and each $\pi (g_h) \in B_h$ because $\pi$ is a graded homomorphism. Because the graded subspaces of $B$ are linearly independent, it follows that each $\pi (g_h) = 0$. Since $g \not = 0$, there exists $k \in F$ such that $g_{k} \not = 0$. By Lemma 3.5, we can write $g_{k}$ as $\sum _{V \in K} a_V 1_{V}$ where $K$ is a finite set of mutually disjoint elements of $B^{\mathrm{co }}_{k}(G)$. Note that $g_k^* = \sum _{V \in K} \overline{a_V} 1_{V^{-1}}$; define $f := g_k^* * g_k$. We claim that $f \in A_e \setminus \{0\}$ and $\pi (f) = 0$. To see this, first notice that

$$\begin{aligned} f = \left( \sum _{V \in K} \overline{a_V} 1_{V^{-1}}\right) * \left( \sum _{W \in K} a_W 1_{W}\right)&= \sum _{V,W \in K} \overline{a_V}a_W 1_{V^{-1}} * 1_{W}\\&= \sum _{V,W \in K} \overline{a_V}a_W 1_{V^{-1}W} \end{aligned}$$

by [22, Proposition 4.5 (3)]. Now, because each $V \in K$ is a subset of $G_{k}$, each $V^{-1}W \subseteq G_{k^{-1} k} = G_e$, and thus $f \in A_e$ as claimed. We have $\pi (f) = 0$ because $\pi (g_{k}) = 0$.

To show that $f$ is nonzero, fix $\alpha \in G_k$ such that $g(\alpha ) \not = 0$. Since the elements of $K$ are mutually disjoint, there is a unique $V_\alpha \in K$ such that $\alpha \in V_\alpha$, and then $a_{V_\alpha } = g(\alpha ) \not = 0$. Since $s$ is a local homeomorphism, $Gs(\alpha )$ is a discrete space. Write $C_c(G s(\alpha ))$ for the space of finitely supported functions from $G s(\alpha )$ to $\mathbb {C}$ and for each $\beta \in Gs(\alpha )$ let $\delta _\beta$ denote the point-mass at $\beta$ so that $C_c(Gs(\alpha )) = \mathrm{span }\{\delta _\beta : \beta \in Gs(\alpha )\}$. For $f \in C_c(G)$, let $\rho (f)$ be the linear map on $C_c(G s(\alpha ))$ determined by

$$\begin{aligned} \rho (f) \delta _{\beta } = \sum _{s(\alpha ) = r(\beta )} f(\alpha ) \delta _{\alpha \beta }. \end{aligned}$$

Let $(\cdot \,|\,\cdot )$ be the standard inner product on $C_c(G s(\alpha ))$, that is $(f|g) = \sum _\beta \overline{f(\beta )} g(\beta )$. Since the elements of $K$ are mutually disjoint, $\big (\rho (1_V) \delta _{s(\alpha )} \mid \rho (1_W) \delta _{s(\alpha )}\big ) = 0$ for distinct $V,W \in K$. A calculation shows that for $V \in K$ and $\beta , \gamma \in G s(\alpha )$, we have $(\delta _\beta | \rho (1_{V^{-1}}) \delta _\gamma ) = (\rho (1_V) \delta _\beta | \delta _\gamma )$. Hence

$$\begin{aligned} \big (\rho (f) \delta _{s(\alpha )} \mid \delta _{s(\alpha )}\big )&= \big (\rho (g_k) \delta _{s(\alpha )} \mid \rho (g_k) \delta _{s(\alpha )}\big ) \\&= \sum _{V,W \in K} \overline{a_V} a_W \big (\rho (1_W) \delta _{s(\alpha )} \mid \rho (1_V) \delta _{s(\alpha )}\big )\\&= \sum _{\begin{array}{c} V \in K,\\ s(\alpha ) \in s(V) \end{array}} |a_V|^2 \ge |a_{V_\alpha }|^2. \end{aligned}$$

Hence $\rho (f) \not = 0$ which forces $f \not = 0$.

By hypothesis $\{u\in G^{(0)}: uG_e u=\{u\}\}$ is dense in $G^{(0)}$. By definition, $A_e$ is equal to the space of locally constant, continuous, compactly supported functions on $G_e$, so we may apply Theorem 5.1 to see that $\pi |_{A_{e}} : A_e \rightarrow B$ annihilates $1_K$ for some compact open $K \subseteq G^{(0)}_e = G^{(0)}$.$\square$

Corollary 5.5

Let $\Gamma$ be a discrete group, $G$ a locally compact, Hausdorff, étale groupoid with totally disconnected unit space, and $c : G\rightarrow \Gamma$ a continuous cocycle. Suppose that $\{u \in G^{(0)}: uG_e u = \{u\}\}$ is dense in $G^{(0)}$. Let $B$ be a $\Gamma$-graded complex algebra and let $\{t_U : U \in B^{\mathrm{co }}_{*}(G)\}$ be a representation of $B^{\mathrm{co }}_{*}(G)$ in $B$. Suppose that $t_U \in B_n$ whenever $U \in B^{\mathrm{co }}_{n}(G)$ and that $t_K \not = 0$ for each compact open $K \subseteq G^{(0)}$. Then the homomorphism $\pi : A(G)\rightarrow B$ obtained from Theorem 3.10 is injective.

Proof

Since each $A(G)_n$ is spanned by $\{1_{U} : U \in B^{\mathrm{co }}_{n}(G)\}$, the homomorphism $\pi$ is graded. Since $\pi (1_K) = t_K \not = 0$ for all compact open $K \subseteq G^{(0)}$, it follows from Theorem 5.4 that $\mathrm{ker }(\pi ) = \{0\}$.$\square$

Remark 5.6

Suppose that $G$ is a locally compact, Hausdorff, étale groupoid with totally disconnected unit space such that $\{u \in G^{(0)}: uGu = \{u\}\}$ is dense in $G^{(0)}$. We may apply Corollary 5.5 with $c$ the trivial cocycle to prove that $A(G)$ is the unique algebra generated by nonzero elements $\{t_U : U \text { is a compact open bisection of }G\}$ satisfying

• (1) $t_\emptyset = 0$;

• (2) $t_Ut_V = t_{UV}$ for all compact open bisections $U,V$; and

• (3) $t_U + t_V = t_{U \cup V}$ whenever $U$ and $V$ are disjoint compact open bisections whose union is a bisection.

Remark 5.7

In the proof of Theorem 5.4, to see that the function $g_k^* * g_k$ was nonzero, we really just checked that its image under Renault’s left-regular representation of $G$ associated to the unit $s(\alpha )$ is nonzero. However, since we are not working in a $C^*$-completion, we can do everything at the level of linear algebra rather than on Hilbert space. We could instead have appealed to the $C^*$-identity by regarding $A(G)$ as a subalgebra of $C_r(G)$, but chose a more elementary argument: our argument is essentially that used by Renault to show that the reduced norm is positive definite on $C_c(G)$.

Remark 5.8

Recall from [8] that if $\Lambda$ is a finitely aligned $k$-graph, then the $k$-graph groupoid $G_\Lambda$ is totally disconnected and locally compact, and carries a $\mathbb {Z}^k$-grading such that ${\{u \in G^{(0)}: uG_e u = \{u\}\}}$ is dense in $G^{(0)}$. So our graded uniqueness theorem applies to $A(G_\Lambda )$ for any finitely aligned $k$-graph. Likewise, Remark 5.6 suggests a Cuntz–Krieger uniqueness theorem for $A(G_\Lambda )$. But in practise the relations described in Definition 3.9 and Remark 5.6 are much harder to verify than those of [3, Definition 3.1].

We do not, at this stage, have any invariants at our disposal to decide whether, given groupoids $G$ and $G'$ satisfying our hypotheses, the algebras $A(G)$ and $A(G')$ are or are not isomorphic. It would be very interesting to develop computable algebraic invariants of $A(G)$ for this purpose, but it is beyond the scope of this paper.

However, as an indication that our construction is more flexible the construction of Kumjian–Pask algebras in [3], we describe a class of groupoids that satisfy our hypotheses but do not obviously arise from $k$-graphs.

Example 5.9

Let $T:X\rightarrow X$ be a surjective local homeomorphism of a totally disconnected, compact, Hausdorff space $X$. Define $T^0 := \mathrm{id }$ and for $k \ge 2$ let $T^k := T \circ \cdots \circ T$ be the $k$-fold self-composite of $T$. Let $G$ be the Deaconu-Renault groupoid defined in [9, Sect. 3]. So

$$\begin{aligned} G= \{(x,n,y)\in X\times \mathbb {Z}\times X\;:\; T^{k}(x)=T^{l}(y),n=k-l\}. \end{aligned}$$

Let $G^{(0)}$ be the subset $\{(x,0,x) : x \in X\}$, which we identify with $X$ in the obvious way. The range and source maps are given by $r(x,n,y) = x$ and $s(x,n,y) = y$. Hence triples $(x_{1},n_{1},y_{1})$ and $(x_{2},n_{2},y_{2})$ are composable if and only if $x_{2}=y_{1}$, in which case $(x_{1},n_{1},y_{1})(x_{2},n_{2},y_{2}) := (x_{1},n_{1}+n_{2},y_{2})$. The inverse of $(x,n,y)$ is $(y,-n,x)$. For open subsets $U,V \subseteq X$ and $k,l \ge 0$ such that $T^{k}|_U$ and $T^{l}|_V$ are homeomorphisms and $T^{k}(U)=T^{l}(V)$, define

$$\begin{aligned} Z(U,V,k,l):=\{(x,k-l,y)\in G:x\in U,y\in V\}. \end{aligned}$$

Then

$$\begin{aligned} \{Z(U,V,k,l) : U,{}&V \subseteq X\text { are compact open}, k,l \ge 0,\\&T^{k}|_U\text { and }T^{l}|_V\text { are homeomorphisms and } T^{k}(U)=T^{l}(V)\} \end{aligned}$$

is a basis of compact open sets for a topology on $G$ under which it becomes a locally compact, Hausdorff groupoid with totally disconnected unit space $X$. Fix $(x, n, y) \in G$ and $k,l$ such that $k - l = n$ and $T^k(x) = T^l(y)$. The source map on $G$ restricts to a homeomorphism on each basic open set $Z(U,V,k,l)$ so is a local homeomorphism. Moreover, the map $c : G\rightarrow \mathbb {Z}$ defined by $c((x,n,y)) = n$ is a cocycle and is continuous because each basic open set belongs to some $c^{-1}(n)$. Hence $(G, c)$ satisfies our hypotheses, and $A(G)$ is a sensible candidate for the Leavitt algebra of $(X, T)$.