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Left regular representations of Garside categories I. C*-algebras and groupoids

Published online by Cambridge University Press:  25 April 2022

Xin Li*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK e-mail: Xin.Li@glasgow.ac.uk
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Abstract

We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

1. Introduction

C*-algebras generated by partial isometries form a rich class of examples, including C*-algebras attached to shifts of finite type [Reference Cuntz12Reference Cuntz14], graph C*-algebras [Reference Raeburn58], higher rank graph C*-algebras [Reference Kumjian and Pask36], C*-algebras attached to self-similiar groups [Reference Nekrashevych49], and semigroup C*-algebras [Reference Coburn9, Reference Cuntz, Echterhoff, Li and Yu15, Reference Li41, Reference Li42]. For instance, it was shown in [Reference Exel and Pardo26] that every UCT Kirchberg algebra arises in this way. The class of UCT Kirchberg algebras plays an important role in the Elliott classification programme for C*-algebras (see [Reference Kirchberg34, Reference Kirchberg and Phillips35, Reference Phillips56, Reference Rørdam64]). Spielberg observed that all the classes of C*-algebras mentioned above can be viewed as special cases of a general, unifying construction of C*-algebras generated by left regular representations of left cancellative small categories [Reference Spielberg68, Reference Spielberg69]. This is a very general construction, as it contains, up to Morita equivalence, all inverse semigroup C*-algebras (see [Reference Donsig, Gensler, King, Milan and Wdowinski21]). These C*-algebras come with a distinguished quotient which is called the boundary quotient. The passage from the C*-algebra to its boundary quotient is analogous to the passage from the Toeplitz-type C*-algebra of a shift of finite type or graph to its Cuntz–Krieger-type C*-algebra.

A powerful way to study these C*-algebras of small categories is to construct a groupoid model and study properties of the C*-algebra through a detailed analysis of the groupoid [Reference Renault62, Reference Spielberg68, Reference Spielberg69]. Actually, there are two candidates for such groupoid models, which both arise from actions of an inverse semigroup on a space of certain filters attached to the small category. The inverse semigroup is given by the left inverse hull, i.e., the smallest inverse semigroup of partial bijections of the small category containing all left multiplication maps by individual elements of the small category. In [Reference Spielberg69], a refined (and enlarged) version of the left inverse hull is considered, leading to the second groupoid model. In both cases, the filters which give rise to the unit space of the groupoid models are defined on the semilattice of idempotents of the inverse semigroup and take into account that elements of this semilattice are subsets of the original small category. The language of inverse semigroups provides an interpretation of the distinguished boundary quotient as the tight quotient, which is induced from the subspace of tight filters (see [Reference Exel23Reference Exel and Pardo25]).

It is an interesting observation that in this very general framework, every left cancellative small category generates – in an entirely natural and intrinsic way – a dynamical system in terms of an inverse semigroup action or a groupoid. The same statement applies to the even more general setting of 0-left cancellative semigroups as considered by Exel and Steinberg [Reference Exel and Steinberg27Reference Exel and Steinberg30]. Generally speaking, the goal would be to find a dictionary between properties of the small category, properties of the inverse semigroup action or groupoid, and properties of the C*-algebra and its boundary quotient. Indeed, we present criteria in terms of the underlying small category which completely characterise when the boundary groupoids – which model the boundary quotients – are Hausdorff, minimal, or effective (or topologically free). We also establish a sufficient criterion for the boundary groupoid to be locally contractive. These properties have immediate consequences for the corresponding boundary quotient C*-algebras concerning ideal structure and pure infiniteness. Such criteria have been established in the general context of inverse semigroup actions and tight groupoids attached to inverse semigroups in [Reference Exel and Pardo25], and it turns out to be fruitful to translate between the work in [Reference Exel and Pardo25] and our setting of small categories. For instance, this leads to generalisations of the results in [Reference Ortega and Pardo53], which covers classes of finitely aligned small categories. In the special case of submonoids of groups, we are naturally led to the following characterisation of topological freeness of the boundary action:

Theorem A. Let P be a submonoid of a group G and denote by $G \curvearrowright \partial \Omega$ its boundary action (in the sense of [Reference Cuntz, Echterhoff, Li and Yu15, Definition 5.7.8]). Define $G^c \mathrel{:=} \left\{ g \in G \text{: } (pP) \cap (gpP) \neq \emptyset \quad \forall \ p \in P \right\}$ .

Then $G \curvearrowright \partial \Omega$ is topologically free if and only if $G^c$ is the trivial group. In this case, $\partial C^*_{\lambda}(P)$ is simple, and $\partial C^*_{\lambda}(P)$ is purely infinite simple unless P is the trivial monoid.

$G^c$ is always a subgroup of G. Theorem A tells us that this subgroup captures topological freeness of the boundary action in an arguably more efficient way than the ‘core’ as in [Reference Crisp and Laca11] (see also [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.7]). In this form, with $G^c$ as the key ingredient, our characterisation of topological freeness of the boundary action has not appeared before, but, as Marcelo Laca and Camila F. Sehnem kindly informed me, it also follows from [Reference Laca and Sehnem40, Proposition 6.18]. We give a self-contained (and short) proof of Theorem A in Section 5 (see Theorem 5.23).

At the same time, our study of boundary groupoids arising from left regular representations of small categories led us to a characterisation of topological freeness of tight groupoids attached to general inverse semigroups (see Theorem 5.11). To the best of the author’s knowledge, such a characterisation was not known before.

We also clarify the relationship between the different groupoid models mentioned above and the analogous variations of the boundary groupoids. For the groupoids themselves, while minimality and local contractiveness are rather rare phenomena, we succeed in completely characterising, in terms of the underlying small category, when the groupoids are Hausdorff or effective (or topologically free). Our criterion for topological freeness is inspired by [Reference Laca and Sehnem40, Theorem 5.9], which treats the special case of submonoids of groups. Furthermore, we establish a characterisation when the boundary is the smallest non-empty closed invariant subspace of the character space, and determine in this case when the boundary groupoid is purely infinite (see Proposition 5.21).

Having identified a natural and unifying general framework, it is important to find classes of small categories which are general enough so that they cover interesting classes of examples and yet concrete enough so that a detailed analysis is possible.

The main goal of the present paper is to discuss one such class of small categories called Garside categories, and in this way contribute to our understanding of C*-algebras attached to small categories. The idea behind Garside categories originated from the study of Braid groups and monoids, and of the more general Artin–Tits groups and monoids. Roughly speaking, Garside structures allow us to carry over classical results and methods from Braid groups and monoids to more general groups, monoids or small categories. The concept of Garside categories feature in proofs of the $K(\pi,1)$ -conjecture for various classes of groups [Reference Bessis4, Reference Paolini and Salvetti54, Reference Paris55]. Recently, a connection has been discovered between Garside categories and Helly graphs, which has several applications, for instance to isomorphism conjectures such as the Farrell–Jones conjecture or the coarse Baum–Connes conjecture [Reference Huang and Osajda32]. We refer the reader to [Reference Dehornoy18] for more details on Garside categories.

In our context, Garside structures allow us to establish normal forms for filters which form the unit spaces of our groupoids. This in turn leads to very concrete descriptions of the groupoid models themselves. As a result, we succeed in describing all closed invariant subspaces in terms of the underlying small category.

Theorem B. Let $\mathfrak{C}$ be a finitely aligned, left cancellative, countable small category and $\mathfrak{S}$ a Garside family in $\mathfrak{C}$ with $\mathfrak{S} \cap \mathfrak{C}^* = \emptyset$ which is $=^*$ -transverse and locally bounded. Let $I_l \ltimes \Omega$ be the groupoid model for $C^*_{\lambda}(\mathfrak{C})$ .

There is a one-to-one correspondence between closed invariant subspaces of $I_l \ltimes \Omega$ and admissible, H-invariant, $\max_{\preceq}^{\infty}$ -closed pairs $(\mathfrak{T},\mathfrak{D})$ with $\mathfrak{T} \subseteq \mathfrak{S}$ and $\mathfrak{D} \subseteq \mathfrak{C}^0$ .

The reader will find more explanations and details in Section 6 (see Theorem 6.25). The point is that our description is purely in terms of the Garside family $\mathfrak{S}$ . We also explicitly characterise which of these closed invariant subspaces belong to the boundary. In addition, we establish criteria for topological freeness and local contractiveness. Again, these properties have consequences for ideal structure and pure infiniteness of our C*-algebras. Our analysis is made possible by the key property of Garside categories that every element admits a normal form, generalising the classical normal form (also called greedy, Garside or Thurston normal form) of elements in Braid and Artin–Tits monoids. Indeed, as explained in [Reference Dehornoy18], the general notion of Garside categories (as in [Reference Dehornoy18]) has been designed to allow for this kind of normal forms. For the purpose of studying groupoids and C*-algebras, the usefulness of normal forms has been observed already, for instance in the context of semigroup C*-algebras of right-angled or spherical Artin–Tits monoids [Reference Crisp and Laca10, Reference Crisp and Laca11, Reference Li, Omland and Spielberg46], or of Baumslag–Solitar monoids [Reference Spielberg67].

As particular examples, we discuss higher rank graphs in Section 7.1. Actually, the starting point for this paper was the observation that every higher rank graph is a Garside category in a very natural way. Our results lead to a new interpretation of gauge-invariant ideals (see Lemma 7.5). Moreover, not only do our results cover the C*-algebras of higher rank graphs, but they also treat Toeplitz algebras. Furthermore, our analysis extends to categories arising from self-similar actions on graphs or higher rank graphs. As another class of concrete examples, we discuss general Artin–Tits monoids. We complete the study of the ideal structure of their semigroup C*-algebras, which has been started in [Reference Crisp and Laca10, Reference Crisp and Laca11, Reference Li, Omland and Spielberg46], by proving the following result:

Theorem C. Let P be an irreducible Artin–Tits monoid with set of atoms A. If P is spherical, then $\textrm{Ker}_{\partial} = \mathcal{K}(\ell^2 P)$ if $\# A = 1$ and $\mathcal{K}(\ell^2 P)$ is the only non-trivial ideal of $\textrm{Ker}_{\partial}$ if $2 \leq \#A < \infty$ . In the latter case, $\textrm{Ker}_{\partial} / \mathcal{K}(\ell^2 P)$ is purely infinite simple. If P is not finitely generated and left reversible, then $\textrm{Ker}_{\partial}$ is purely infinite simple. If P is finitely generated and not spherical, then $\mathcal{K}(\ell^2 P)$ is the only non-trivial ideal of $C^*_{\lambda}(P)$ , and $C^*_{\lambda}(P) / \mathcal{K}(\ell^2 P)$ is purely infinite simple. If P is not finitely generated and not left reversible, then $C^*_{\lambda}(P)$ is purely infinite simple.

Here, $\textrm{Ker}_{\partial}$ is the kernel of the canoncial projection $C^*_{\lambda}(P) \twoheadrightarrow \partial C^*_{\lambda}(P)$ . In the spherical or left reversible case, $\partial C^*_{\lambda}(P)$ coincides with the reduced group C*-algebra of the Artin–Tits group corresponding to P. In Theorem C, the finitely generated, spherical case is treated in [Reference Li, Omland and Spielberg46], and the right-angled case is treated in [Reference Crisp and Laca10, Reference Crisp and Laca11]. Our contribution concerns the remaining cases. We can also characterise when $C^*_{\lambda}(P)$ or $\textrm{Ker}_{\partial}$ is nuclear (see also [Reference Laca and Li38, Theorem 4.2]). Moreover, we point out that K-theory for semigroup C*-algebras of Artin–Tits monoids has been computed in [Reference Li44], assuming that the corresponding Artin–Tits group satisfies the Baum–Connes conjecture with coefficients.

Higher rank graphs and Artin–Tits monoids are just some examples of Garside categories. The reader will find many more examples in [Reference Dehornoy18].

Apart from providing a natural class of examples where we can test and develop our understanding of C*-algebras attached to small categories, this paper at the same time sets the stage for a detailed analysis of the groupoids arising from left regular representations of small categories. These groupoids are not only auxiliary structures to translate between small categories and their C*-algebras, but they are also interesting on their own right as they lead to interesting new structures, for instance topological full groups. Our original motivation which led to the present paper was the natural question left open by Matui in [Reference Matui48, Section 5.3] whether topological full groups of groupoids attached to products of shifts of finite type are of type $\textrm{F}_{\infty}$ . We answer this question in [Reference Li45].

2. Preliminaries

Let us recall some basics regarding left regular representations of left cancellative categories, C*-algebras generated by these representations and groupoid models for these C*-algebras. Note that we view categories – which will all be assumed to be small in this paper – as generalisations of monoids (as in [Reference Witzel70]), so that no sophisticated category theory will be used.

2.1. Left cancellative small categories, their left regular representations and C*-algebras

Given a small category with set of morphisms $\mathfrak{C}$ , let $\mathfrak{C}^0$ be its set of objects. We will identify $\mathfrak{v} \in \mathfrak{C}^0$ with the identity morphism at $\mathfrak{v}$ , so that $\mathfrak{C}^0$ is identified with a subset of $\mathfrak{C}$ . Often, we will abuse notation and simply call $\mathfrak{C}$ the small category. Let $\mathfrak{d}: \: \mathfrak{C} \to \mathfrak{C}^0$ and $\mathfrak{t}: \: \mathfrak{C} \to \mathfrak{C}^0$ be the domain and target maps, so that for $c, d \in \mathfrak{C}$ , the product cd is defined if and only if $\mathfrak{d}(c) = \mathfrak{t}(d)$ . This means that our convention is the same as the one in [Reference Spielberg69, Reference Witzel70], while it is opposite to the one used in [Reference Dehornoy18] (see [Reference Witzel70, Remark 1.1]). For $c \in \mathfrak{C}$ and $S \subseteq \mathfrak{C}$ , we set $cS \mathrel{:=} \left\{ cs \text{: } s \in S, \, \mathfrak{t}(s) = \mathfrak{d}(c) \right\}$ . Moreover, $\mathfrak{C}^*$ denotes the set of invertible elements of $\mathfrak{C}$ , i.e., elements $c \in \mathfrak{C}$ for which there exists $c^{-1} \in \mathfrak{C}$ with $c^{-1} c = \mathfrak{d}(c)$ and $c c^{-1} = \mathfrak{t}(c)$ . Note that $\mathfrak{C}^*$ is denoted by $\mathfrak{C}^\times$ in [Reference Dehornoy18, Reference Witzel70].

Definition 2.1. A small category $\mathfrak{C}$ is called left cancellative if for all $c, x, y \in \mathfrak{C}$ with $\mathfrak{d}(c) = \mathfrak{t}(x) = \mathfrak{t}(y)$ , $cx = cy$ implies $x = y$ .

From now on, all our small categories will be assumed to be left cancellative. Let $\mathfrak{C}$ be such a small category and form the Hilbert space $\ell^2 \mathfrak{C}$ , with canonical orthonormal basis given by $\delta_x(y) = 1$ if $x=y$ and $\delta_x(y) = 0$ if $x \neq y$ . For each $c \in \mathfrak{C}$ , the assignment $\delta_x \mapsto \delta_{cx}$ if $\mathfrak{t}(x) = \mathfrak{d}(c)$ and $\delta_x \mapsto 0$ if $\mathfrak{t}(x) \neq \mathfrak{d}(c)$ extends to a bounded linear operator on $\ell^2 \mathfrak{C}$ which we denote by $\lambda_c$ . Note that it is at this point, i.e., to ensure boundedness, that we need left cancellation, which actually implies that $\lambda_c$ is a partial isometry. The left regular representation of $\mathfrak{C}$ is given by $\mathfrak{C} \to \textrm{PIsom}(\ell^2 \mathfrak{C}), \, c \mapsto \lambda_c$ , where PIsom stands for the set of partial isometries.

Definition 2.2. The left reduced C*-algebra of $\mathfrak{C}$ is given by $C^*_{\lambda}(\mathfrak{C}) \mathrel{:=} C^*(\left\{ \lambda_c \text{: } c \in \mathfrak{C} \right\}) \subseteq \mathcal{L}(\ell^2 \mathfrak{C})$ .

2.2. Inverse semigroup actions and groupoid models

Let us now describe (candidates for) groupoid models for $C^{*}_{\lambda}(\mathfrak{C})$ . First of all, every $c \in \mathfrak{C}$ induces the partial bijection $\mathfrak{d}(c) \mathfrak{C} \xrightarrow{\sim} c \mathfrak{C}, \, x \mapsto cx$ . For brevity, we denote this partial bijection by c again.

Definition 2.3. The left inverse hull $I_l$ of $\mathfrak{C}$ is the smallest inverse semigroup containing the partial bijections $\left\{ c \text{: } c \in \mathfrak{C} \right\}$ , i.e., the smallest semigroup of partial bijections of $\mathfrak{C}$ containing the partial bijections $\left\{ c \text{: } c \in \mathfrak{C} \right\}$ and closed under inverses.

For more details on inverse semigroups, we refer the reader to [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.5.1]. For $s \in I_l$ , we denote its domain by $\textrm{dom}\ (s)$ and its image by $\textrm{im}\ (s)$ . Following [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.5.1], in case $I_l$ contains the partial bijection 0 which is nowhere defined, $\emptyset \xrightarrow{\sim} \emptyset$ , we say that $I_l$ contains zero, and we view $I_l$ as an inverse semigroup with zero. A typical nonzero element $s \in I_l$ is of the form $s = d_n^{-1} c_n \dotso d_1^{-1} c_1$ for some $d_i, c_i \in \mathfrak{C}$ with $\mathfrak{t}(c_i) = \mathfrak{t}(d_i)$ and $\mathfrak{d}(d_i) = \mathfrak{d}(c_{i+1})$ .

Remark 2.4. Elements of $I_l$ are called zigzags in [Reference Spielberg69].

Definition 2.5. For $0 \neq s \in I_l$ , define $\mathfrak{d}(s)$ as the unique $\mathfrak{v} \in \mathfrak{C}^0$ such that $\textrm{dom}\ (s) \subseteq \mathfrak{v} \mathfrak{C}$ , and define $\mathfrak{t}(s)$ as the unique $\mathfrak{w} \in \mathfrak{C}^0$ such that $\textrm{im}\ (s) \subseteq \mathfrak{w} \mathfrak{C}$ .

Such $\mathfrak{v}$ and $\mathfrak{w}$ exist because, if $s = d_n^{-1} c_n \dotso d_1^{-1} c_1$ , then $\textrm{dom}\ (s) \subseteq \textrm{dom}\ (c_1) \subseteq \mathfrak{d}(c_1) \mathfrak{C}$ and $\textrm{im}\ (s) \subseteq \textrm{im}\ (d_n^{-1}) \subseteq \mathfrak{d}(d_n) \mathfrak{C}$ .

Definition 2.6. The semilattice of idempotents of $I_l$ is denoted by $\mathcal{J} \mathrel{:=} \left\{ s^{-1}s \text{: } s \in I_l \right\} = \left\{ ss^{-1} \text{: } s \in I_l \right\}$ .

$I_l$ contains 0 if and only if $\mathcal{J}$ contains $\emptyset$ . In that case we denote $\emptyset \in \mathcal{J}$ by 0 again.

Alternatively, we could set $\mathcal{J} = \left\{ \textrm{dom}\ (s) \text{: } s \in I_l \right\} = \left\{ \textrm{im}\ (s) \text{: } s \in I_l \right\}$ . $\mathcal{J}$ is the analogue of the set of constructible right ideals in the semigroup context (see [Reference Li41]). Multiplication in $\mathcal{J}$ (denoted by ef for $e, f \in \mathcal{J}$ ) corresponds to intersection of subsets of $\mathfrak{C}$ , and the partial order “ $\leq$ ” on $\mathcal{J}$ corresponds to inclusion of subsets.

At this point, we present a variation of $I_l$ , following [Reference Spielberg69].

Definition 2.7. Let $\bar{\mathcal{J}}$ denote the set of subsets of $\mathfrak{C}$ of the form $e \setminus \bigcup_{i=1}^n f_n$ for some $e, f_1, \dotsc, f_n \in \mathcal{J}$ with $f_1, \dotsc, f_n \leq e$ .

Let $\bar{I}_l$ be the set of all partial bijections of $\mathfrak{C}$ of the form $s \varepsilon$ for $s \in I_l$ and $\varepsilon \in \bar{\mathcal{J}}$ with $\varepsilon \leq s^{-1}s$ .

It is easy to see that $\bar{I}_l$ is again an inverse semigroup, whose semilattice of idempotents is given by $\bar{\mathcal{J}}$ .

Definition 2.8. The space of characters $\widehat{\mathcal{J}}$ is given by the set of non-zero multiplicative maps $\mathcal{J} \to \left\{ 0,1 \right\}$ , which send $0 \in \mathcal{J}$ to $0 \in \left\{ 0,1 \right\}$ in case $I_l$ contains 0. Here multiplication in $\left\{ 0,1 \right\}$ is the usual one induced by multiplication in $\mathbb{R}$ . The topology on $\widehat{\mathcal{J}}$ is given by point-wise convergence.

A basis of compact open sets for the topology of $\widehat{\mathcal{J}}$ is given by sets of the form

\begin{equation*} \widehat{\mathcal{J}}(e;\mathfrak{f}) \mathrel{:=} \big \lbrace \chi \in \widehat{\mathcal{J}}: \: \chi(e) = 1, \, \chi(f) = 0 \ \ \forall \, f \in \mathfrak{f} \big \rbrace,\end{equation*}

where $e \in \mathcal{J}$ and $\mathfrak{f} \subseteq \mathcal{J}$ is a finite subset. By replacing $\mathfrak{f}$ by $\left\{ ef \text{: } f \in \mathfrak{f} \right\}$ , we can always arrange that $f \leq e$ for all $f \in \mathfrak{f}$ . We will also set $\widehat{\mathcal{J}}(e) \mathrel{:=} \lbrace \chi \in \widehat{\mathcal{J}}: \: \chi(e) = 1 \rbrace$ . Since $\mathfrak{v} \mathfrak{C} \cap \mathfrak{w} \mathfrak{C} = \emptyset$ if $\mathfrak{v} \neq \mathfrak{w}$ , for every $\chi \in \widehat{\mathcal{J}}$ there exists a unique $\mathfrak{v} \in \mathfrak{C}^0$ with $\chi(\mathfrak{v} \mathfrak{C}^0) = 1$ . In other words, we have $ \widehat{\mathcal{J}} = \coprod_{\mathfrak{v} \in \mathfrak{C}^0} \widehat{\mathcal{J}}(\mathfrak{v})$ . As explained in [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.5.1], there is a one-to-one correspondence between elements in $\widehat{\mathcal{J}}$ and filters (on $\mathcal{J}$ ), i.e., nonempty subsets $\mathcal{F}$ of $\mathcal{J}$ with the properties that $0 \notin \mathcal{F}$ if $I_l$ contains 0, whenever $e, f \in \mathcal{J}$ satisfy $e \leq f$ , then $e \in \mathcal{F}$ implies $f \in \mathcal{F}$ , and whenever $e, f \in \mathcal{J}$ lie in $\mathcal{F}$ , then ef must lie in $\mathcal{F}$ as well. To be concrete, the one-to-one correspondence is implemented by $\widehat{\mathcal{J}} \ni \chi \mapsto \chi^{-1}(1) \subseteq \mathcal{J}$ .

Following [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.6.7] and [Reference Spielberg69], we now construct a subspace of $\widehat{\mathcal{J}}$ which takes into account that elements of $\mathcal{J}$ are subsets of $\mathfrak{C}$ . First, let $D_{\lambda}(\mathfrak{C}) \mathrel{:=} \overline{\textrm{span}}(\left\{ 1_e \text{: } e \in \mathcal{J} \right\}) \subseteq \ell^{\infty}(\mathfrak{C})$ . Here $1_e$ denotes the characteristic function of $e \subseteq \mathfrak{C}$ . As explained in [Reference Cuntz, Echterhoff, Li and Yu15, Corollary 5.6.28], the spectrum of $D_{\lambda}(\mathfrak{C})$ can be identified with the following subspace of $\widehat{\mathcal{J}}$ :

Definition 2.9. Let $\Omega$ be the subspace of $\widehat{\mathcal{J}}$ consisting of characters $\chi$ with the property that whenever $e, f_1, \dotsc, f_n \in \mathcal{J}$ satisfy $e = \bigcup_{i=1}^n f_i$ as subsets of $\mathfrak{C}$ , then $\chi(e) = 1$ implies that $\chi(f_i) = 1$ for some $1 \leq i \leq n$ .

Remark 2.10. Following [Reference Cuntz, Echterhoff, Li and Yu15, Corollary 5.6.28], we will view every $\chi \in \Omega$ as a character on $D_{\lambda}(\mathfrak{C})$ , again denoted by $\chi$ . Given $\varepsilon = e \setminus \bigcup_{i=1}^n f_n \in \bar{\mathcal{J}}$ , we have $1_{\varepsilon} \in D_{\lambda}(\mathfrak{C})$ , and we set $\chi(\varepsilon) \mathrel{:=} \chi(1_{\varepsilon})$ .

Example 2.11. Given $x \in \mathfrak{C}$ , define $\chi_x(e) \mathrel{:=} 1$ if $x \mathfrak{C} \leq e$ and $\chi_x(e) \mathrel{:=} 0$ if $x \mathfrak{C} \not\leq e$ . It is easy to see that $\chi_x \in \Omega$ .

The following is immediate from the definition of the topology of $\Omega$ , using the basis of compact open sets as defined above.

Lemma 2.12. $\left\{ \chi_x \text{: } x \in \mathfrak{C} \right\}$ is a dense subset of $\Omega$ .

The following observation is an immediate consequence of [Reference Cuntz, Echterhoff, Li and Yu15, Corollary 5.6.29].

Lemma 2.13. We have $\Omega = \widehat{\mathcal{J}}$ if and only if whenever $e, f_1, \dotsc, f_n \in \mathcal{J}$ satisfy $e = \bigcup_{i=1}^n f_i$ as subsets of $\mathfrak{C}$ , then there exists $1 \leq i \leq n$ with $e = f_i$ .

Let us now dualise and obtain the following action of $I_l$ on $\widehat{\mathcal{J}}$ . A given $s \in I_l$ induces the partial homeomorphism $\widehat{\mathcal{J}}(s^{-1}s) \xrightarrow{\sim} \widehat{\mathcal{J}}(ss^{-1}), \, \chi \mapsto s.\chi \mathrel{:=} \chi(s^{-1} \sqcup s)$ . These partial homeomorphism give rise to an action $I_l \curvearrowright \widehat{\mathcal{J}}$ . The same proof as for [Reference Cuntz, Echterhoff, Li and Yu15, Lemma 5.6.40] shows that $\Omega$ is $I_l$ -invariant, so that we obtain an $I_l$ -action $I_l \curvearrowright \Omega$ by restriction. As before, a given $s \in I_l$ acts via the partial homeomorphism $\Omega(s^{-1}s) \xrightarrow{\sim} \Omega(ss^{-1}), \, \chi \mapsto \chi(s^{-1} \sqcup s)$ . Here and in the sequel, given a subspace $X \subseteq \widehat{\mathcal{J}}$ , we set $X(e) \mathrel{:=} X \cap \widehat{\mathcal{J}}(e)$ and $X(e;\mathfrak{f}) \mathrel{:=} X \cap \widehat{\mathcal{J}}(e;\mathfrak{f})$ .

We now set out to describe two candidates for a groupoid model for $C^*_{\lambda}(\mathfrak{C})$ . First, we set

\begin{equation*} I_l * \Omega \mathrel{:=} \left\{ (s,\chi) \in I_l \times \Omega \text{: } \chi(s^{-1}s) = 1 \right\}.\end{equation*}

Definition 2.14. The transformation groupoid $I_l \ltimes \Omega$ is given by $I_l * \Omega / { }_{\sim}$ , where we set $(s,\chi) \sim (t,\psi)$ if $\chi = \psi$ and there exists $e \in \mathcal{J}$ with $\chi(e) = 1$ and $se = te$ . Equivalence classes with respect to $\sim$ are denoted by $[\cdot]$ , and for $s \in I_l$ and $U \subseteq \Omega$ , we set $[s,U] \mathrel{:=} \left\{ [s,\chi] \text{: } \chi \in U \right\}$ . Range and source maps are given by ${\textrm{r}}([s,\chi]) = s.\chi$ and ${\textrm{s}}([s,\chi]) = \chi$ . Multiplication and inversion are defined by $[s,t.\chi][t,\chi] = [st,\chi]$ and $[s,\chi]^{-1} = [s^{-1},s.\chi]$ .

We equip $I_l \ltimes \Omega$ with the unique topology such that for all $s \in I_l$ , $[s,\Omega(s^{-1}s)]$ is an open subset of $I_l \ltimes \Omega$ and the source map induces a homeomorphism $[s,\Omega(s^{-1}s)] \xrightarrow{\sim} \Omega(s^{-1}s)$ .

As explained in [Reference KwaŚniewski and Meyer37, Section 2.1], we call $I_l \ltimes \Omega$ the transformation groupoid and not the groupoid of germs (as in for instance [Reference Exel and Pardo25]) because in other contexts, the groupoid of germs denotes the quotient of a groupoid by the interior of its isotropy subgroupoid (see for instance [Reference Renault63]).

Now we follow [Reference Spielberg69, Section 5] and construct a variation of $I_l \ltimes \Omega$ .

Definition 2.15. We define $I_l \,{\,{\bar{\ltimes}}\,}\, \Omega \mathrel{:=} I_l * \Omega / { }_{\bar{\sim}}$ , where we set $(s,\chi) \bar{\sim} (t,\psi)$ if $\chi = \psi$ and there exists $\varepsilon \in \bar{\mathcal{J}}$ with $\chi(\varepsilon) = 1$ and $s\varepsilon = t\varepsilon$ in $\bar{I}_l$ . Equivalence classes with respect to $\bar{\sim}$ are denoted by $[\cdot]^{\bar{\sim}}$ . The groupoid structure on $\bar{I}_l \ltimes \Omega$ is defined in the same way as for $I_l \ltimes \Omega$ .

We equip $I_l \,{\bar{\ltimes}}\, \Omega$ with the unique topology such that for all $s \in I_l$ , $[s,\Omega(s^{-1}s)]^{\bar{\sim}}$ is an open subset of $I_l \,{\bar{\ltimes}}\, \Omega$ and the source map induces a homeomorphism $[s,\Omega(s^{-1}s)]^{\bar{\sim}} \xrightarrow{\sim} \Omega(s^{-1}s)$ .

Remark 2.16. It is straightforward to check that the $I_l$ -action on $\Omega$ induces an $\bar{I}_l$ -action $\bar{I}_l \curvearrowright \Omega$ such that the inclusion $I_l \hookrightarrow \bar{I}_l$ induces an isomorphism between the transformation groupoid $\bar{I}_l \ltimes \Omega$ for $\bar{I}_l \curvearrowright \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ given by $l_l \,{\bar{\ltimes}}\, \Omega \xrightarrow{\sim} \bar{I}_l \ltimes \Omega, \, [s,\chi]^{\bar{\sim}} \mapsto [s,\chi]$ .

By construction, we have a canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ . It is easy to see that this projection induces an isomorphism of the groupoids of germs.

2.3. Finite alignment

Let us now introduce a condition which allows us to reduce the discussion from general constructible right ideals to principal right ideals.

Definition 2.17. ([Reference Spielberg69, Definition 3.2]) $\mathfrak{C}$ is finitely aligned if for all $a,b \in \mathfrak{C}$ , there exists a finite subset $F \subseteq \mathfrak{C}$ such that $a \mathfrak{C} \cap b \mathfrak{C} = \bigcup_{c \in F} c \mathfrak{C}$ .

Remark 2.18. The notion of finite alignment is closely related to the notion of minimal common right multiple (see [Reference Dehornoy18, Definition 2.38]), which we abbreviate by mcm. Given $a, b, c \in \mathfrak{C}$ , c is called an mcm if $c \in a \mathfrak{C} \cap b \mathfrak{C}$ and no proper left divisor d (i.e., an element $d \in \mathfrak{C}$ with $c \in d \mathfrak{C}$ ) satisfies $d \in a \mathfrak{C} \cap b \mathfrak{C}$ . It is immediate from [Reference Spielberg69, Lemma 3.3] that $\mathfrak{C}$ is finitely aligned if and only if for all $a, b \in \mathfrak{C}$ , the set of mcms $\textrm{mcm}(a,b)$ is non-empty and finite up to right multiplication by $\mathfrak{C}^*$ .

The following observations are immediate from our definitions (see also [Reference Spielberg69, Section 3]).

Lemma 2.19. Suppose that $\mathfrak{C}$ is finitely aligned. Then the following hold:

  1. (i) For all $e \in \mathcal{J}$ there exists a finite subset $F \subseteq \mathfrak{C}$ such that $e = \bigcup_{x \in F} x \mathfrak{C}$ , and every $\varepsilon \in \bar{\mathcal{J}}$ is a finite disjoint union of sets of the form $x \mathfrak{C} \setminus \bigcup_{i=1}^n y_i \mathfrak{C}$ for $x, y_1, \dotsc, y_n \in \mathfrak{C}$ .

  2. (ii) Every $\chi \in \Omega$ is determined by $\mathcal{F}_\textrm{p} \mathrel{:=} \left\{ x \mathfrak{C} \subseteq \mathfrak{C} \text{: } x \in \mathfrak{C}, \, \chi(x \mathfrak{C}) = 1 \right\}$ , in the sense that for arbitrary $e \in \mathcal{J}$ , $\chi(e) = 1$ if and only if there exists $x \mathfrak{C} \in \mathcal{F}_\textrm{p}$ with $x \mathfrak{C} \leq e$ . Moreover, a basis of compact open sets for $\Omega$ is given by sets of the form $\Omega(x \mathfrak{C}; y_1 \mathfrak{C}, \dotsc, y_n \mathfrak{C})$ .

  3. (iii) Every $s \in I_l$ is a finite union of partial bijections of the form $c d^{-1}$ , where $d,c \in \mathfrak{C}$ satisfy $\mathfrak{d}(c) = \mathfrak{d}(d)$ .

  4. (iv) We have

    \begin{eqnarray*} I_l \ltimes \Omega &=& \left\{ [cd^{-1},\chi] \text{: } c, d \in \mathfrak{C}, \, \mathfrak{d}(c) = \mathfrak{d}(d); \; (cd^{-1},\chi) \in I_l * \Omega \right\},\\[3pt] I_l \,{\bar{\ltimes}}\, \Omega &=& \left\{ [cd^{-1},\chi]^{\bar{\sim}} \text{: } c, d \in \mathfrak{C}, \, \mathfrak{d}(c) = \mathfrak{d}(d); \; (cd^{-1},\chi) \in I_l * \Omega \right\}.\end{eqnarray*}

In this sense, finite alignment allows us to reduce to principal right ideals.

2.4. Groupoid models for left regular C*-algebras

Following [Reference Spielberg69], we now explain in what sense $I_l \,{\bar{\ltimes}}\, \Omega$ is a groupoid model for $C^*_{\lambda}(\mathfrak{C})$ . First of all, as explained in [Reference Spielberg69, Section 11], there is a canonical projection $\Lambda: \: C^*_r(I_l \,{\bar{\ltimes}}\, \Omega) \twoheadrightarrow C^*_{\lambda}(\mathfrak{C})$ given by $\Lambda(1_{[s,\Omega(s^{-1}s)]^{\bar{\sim}}})(\delta_x) = \delta_{s(x)}$ if $x \in \textrm{dom}\ (s)$ and $\Lambda(1_{[s,\Omega(s^{-1}s)]^{\bar{\sim}}})(\delta_x) = 0$ if $x \notin \textrm{dom}\ (s)$ . Moreover, it is shown in [Reference Spielberg69, Section 11] that $\Lambda$ is an isomorphism if $\mathfrak{C}$ is finitely aligned or $I_l \,{\bar{\ltimes}}\, \Omega$ is Hausdorff. We present a characterisation for the Hausdorff property in Lemma 4.1. After comparing the groupoids $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ , we obtain similar results for $I_l \ltimes \Omega$ . The reader will also find examples for which $\Lambda$ fails to be injective in [Reference Spielberg69, Section 11].

2.5. The boundary

Finally, we introduce the boundary, following [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.7].

Definition 2.20. $\widehat{\mathcal{J}}_{\max}$ denotes the set of characters $\chi \in \widehat{\mathcal{J}}$ for which $\chi^{-1}(1)$ is maximal among all characters $\chi \in \widehat{\mathcal{J}}$ .

The same proof as for [Reference Cuntz, Echterhoff, Li and Yu15, Lemma 5.7.7] shows that $\widehat{\mathcal{J}}_{\max} \subseteq \Omega$ . Hence, this justifies the notation $\Omega_{\max} \mathrel{:=} \widehat{\mathcal{J}}_{\max}$ . The following collects observations about $\Omega_{\max}$ , which are proven in the same way as in [Reference Cuntz, Echterhoff, Li and Yu15, Section 5.7].

Lemma 2.21.

  1. (i) If $I_l$ contains 0, then $\chi \in \widehat{\mathcal{J}}$ lies in $\Omega_{\max}$ if and only if for all $e \in \mathcal{J}$ with $\chi(e) = 0$ , there exists $f \in \mathcal{J}$ with $\chi(f) = 1$ such that $ef = 0$ .

  2. (ii) For all $0 \neq e \in \mathcal{J}$ , there exists $\chi \in \Omega_{\max}$ with $\chi(e) = 1$ .

  3. (iii) $\Omega_{\max}$ is $I_l$ -invariant.

Definition 2.22. We define the boundary as $\partial \Omega \mathrel{:=} \overline{\Omega_{\max}} \subseteq \Omega$ .

By Lemma 2.21 (iii), $\partial \Omega$ is $I_l$ -invariant, so that we may form the boundary groupoids.

Definition 2.23. We define the boundary groupoids as $I_l \ltimes \partial \Omega$ and $I_l \,{\bar{\ltimes}}\, \partial \Omega$ .

This also leads to the boundary quotients $C^*_r(I_l \ltimes \partial \Omega)$ and $C^*_r(I_l \,{\bar{\ltimes}}\, \partial \Omega)$ .

Remark 2.24. The boundary groupoid $I_l \ltimes \partial \Omega$ can be identified with the tight groupoid of the left inverse hull $I_l$ , in the sense of [Reference Exel23, Reference Exel and Pardo25]. However, an analogous statement does not hold for $I_l \,{\bar{\ltimes}}\, \partial \Omega$ . Indeed, as noted in [Reference Spielberg69, Section 6], $\widehat{\bar{\mathcal{J}}}_{\max}$ can be identified with $\Omega$ . It follows that $\widehat{\bar{\mathcal{J}}}_{\max} = \partial \widehat{\bar{\mathcal{J}}}$ , i.e., $\widehat{\bar{\mathcal{J}}}_{\max}$ itself is already closed. It is also easy to see this directly. This means that the tight groupoid of the inverse semigroup $\bar{I}_l$ is given by $I_l \,{\bar{\ltimes}}\, \Omega$ . Thus, $I_l \,{\bar{\ltimes}}\, \partial \Omega$ does not have an obvious description as a tight groupoid attached to an inverse semigroup.

3. Comparison of groupoid models

Let us address the natural question when the groupoids $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ are isomorphic. By construction, there is a canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ .

First, we collect a few observations which are immediate consequences of our construction.

Lemma 3.1.

  1. (i) The canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ is an open quotient map.

  2. (ii) The canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ maps bisections to bisections.

  3. (iii) The identity map on $\Omega$ induces a bijection between subsets which are invariant for $I_l \ltimes \Omega$ and subsets which are invariant for $I_l \,{\bar{\ltimes}}\, \Omega$ .

Lemma 3.2. The canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ is an isomorphism if one of the following holds:

  1. (i) $\mathfrak{C}$ is finitely aligned.

  2. (ii) $I_l \ltimes \Omega$ is Hausdorff.

Proof. Take $(s,\chi), (t,\chi) \in I_l * \Omega$ with $(s,\chi) \bar{\sim} (t,\chi)$ . Then there exists $\varepsilon \in \bar{\mathcal{J}}$ with $\chi(\varepsilon) = 1$ and $s \varepsilon = t \varepsilon$ .

Suppose that (i) holds. By Lemma 2.19 (i), we may assume that $\varepsilon = x \mathfrak{C} \setminus \bigcup_{i=1}^n y_i \mathfrak{C}$ for some $x, y_1, \dotsc, y_n \in \mathfrak{C}$ . Then $s \varepsilon = t \varepsilon$ implies $s(x) = t(x)$ , so that, with $e \mathrel{:=} x \mathfrak{C}$ , $s e = t e$ . Moreover, $\chi(\varepsilon) = 1$ implies $\chi(e) = 1$ since $\varepsilon \leq e$ . This shows that $(s,\chi) \sim (t,\chi)$ .

Now assume that (ii) holds. By Lemma 2.12, we can find $x_i \in \mathfrak{C}$ with $\lim_i \chi_{x_i} = \chi$ . As $\chi(\varepsilon) = 1$ , we may assume $\chi_{x_i}(\varepsilon) = 1$ , i.e., $x_i \in \varepsilon$ . Setting $e_i \mathrel{:=} x_i \mathfrak{C}$ , $s \varepsilon = t \varepsilon$ implies $s e_i = t e_i$ , and thus $(s,\chi_{x_i}) \sim (t,\chi_{x_i})$ . Because $\lim_i (s,\chi_{x_i}) = (s,\chi)$ and $\lim_i (t,\chi_{x_i}) = (t,\chi)$ , and since $I_l \ltimes \Omega$ is Hausdorff, we conclude that $(s,\chi) \sim (t,\chi)$ .

For a characterisation of the Hausdorff property for $I_l \ltimes \Omega$ , see Lemma 4.1.

Remark 3.3. As observed in Section 2.2, the canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ induces an isomorphism at the level of groupoids of germs. Hence if $I_l \ltimes \Omega$ is effective, the canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ must be an isomorphism.

The following is an immediate consequence of the results mentioned in Section 2.4 and Lemma 3.2

Corollary 3.4. If $\mathfrak{C}$ is finitely aligned or $I_l \ltimes \Omega$ is Hausdorff, then $C^*_r(I_l \ltimes \Omega)$ is isomorphic to $C^*_{\lambda}(\mathfrak{C})$ .

Let us now compare boundary groupoids.

Lemma 3.5. The canonical projection $I_l \ltimes \partial \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \partial \Omega$ is an isomorphism if one of the following holds:

  1. (i) The canonical projection $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ is an isomorphism.

  2. (ii) $I_l \ltimes \partial \Omega$ is Hausdorff.

  3. (iii) $\partial \Omega = \Omega_{\max}$ .

Proof. It is easy to see that (i) is a sufficient condition. Now take $(s,\chi), (t,\chi) \in I_l * \Omega$ with $(s,\chi) \bar{\sim} (t,\chi)$ . Then there exists $\varepsilon \in \bar{\mathcal{J}}$ with $\chi(\varepsilon) = 1$ and $s \varepsilon = t \varepsilon$ , where $\varepsilon = e \setminus \bigcup_{i=1} f_i$ for $e, f_1, \dotsc, f_n \in \mathcal{J}$ . We first show that if $\chi \in \Omega_{\max}$ , then $(s,\chi) \sim (t,\chi)$ : Indeed, $\chi(\varepsilon) = 1$ implies that $\chi(f_i) = 0$ for all $1 \leq i \leq n$ . By Lemma 2.21 (i), $\chi(f_i) = 0$ implies that there exists $f'_i \in \mathcal{J}$ with $\chi(f'_i) = 1$ and $f_i f'_i = 0$ . Set $f' \mathrel{:=} f'_1 \dotsm f'_n$ . Then $\chi(f') = 1$ and $f' f_i = 0$ for all $1 \leq i \leq n$ . We conclude that $\chi(e f') = 1$ . Moreover, $e f' \subseteq \varepsilon$ , so that $s e f' = t e f'$ . It follows that $(s,\chi) \sim (t,\chi)$ , as desired. This immediately implies that (iii) is a sufficient condition. To treat (ii), assume now that $(s,\chi) \bar{\sim} (t,\chi)$ for some $\chi \in \partial \Omega$ . Then there exist $\chi_i \in \Omega_{\max}$ with $\lim_i \chi_i = \chi$ . We may assume $\chi_i(\varepsilon) = 1$ since $\chi(\varepsilon) = 1$ . It follows that $(s,\chi_i) \bar{\sim} (t,\chi_i)$ , and, by what we just proved, $(s,\chi_i) \sim (t,\chi_i)$ . Since $I_l \ltimes \partial \Omega$ is Hausdorff, we conclude $\lim_i (s,\chi_i) = (s,\chi) \sim (t,\chi) = \lim_i (t,\chi_i)$ , as desired.

Question 3.6. Do we always have isomorphisms $I_l \ltimes \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \Omega$ and $I_l \ltimes \partial \Omega \twoheadrightarrow I_l \,{\bar{\ltimes}}\, \partial \Omega$ ? Most likely the answer will be negative, in which case it would be interesting to find concrete examples where the canonical projections fail to be injective.

4. Properties of the groupoids

We characterise when $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ are Hausdorff, when $I_l \ltimes \Omega$ is topologically free, and when $I_l \,{\bar{\ltimes}}\, \Omega$ is effective. These properties have consequences for the reduced C*-algebras of $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ (see Corollary 4.10).

Let us start with the Hausdorff property. The following will be an application of [Reference Exel and Pardo25, Theorem 3.15].

Lemma 4.1.

  1. (i) $I_l \ltimes \Omega$ is Hausdorff if and only if for all $s \in I_l$ , there exists a (possibly empty) finite subset $\left\{ e_1, \dotsc, e_n \right\} \subseteq \mathcal{J}$ with $ \left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\} = \bigcup_{i=1}^n e_i$ .

  2. (ii) $I_l \,{\bar{\ltimes}}\, \Omega$ is Hausdorff if and only if for all $s \in I_l$ , there exists a (possibly empty) finite subset $\left\{ \varepsilon_1, \dotsc, \varepsilon_n \right\} \subseteq \bar{\mathcal{J}}$ with $ \left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\} = \bigcup_{i=1}^n \varepsilon_i$ .

Proof.

  1. (i) [Reference Exel and Pardo25, Theorem 3.15] implies that $I_l \ltimes \Omega$ is Hausdorff if and only if for all $s \in I_l$ , the subset

    (4.1) \begin{equation} \left\{ \chi \in \Omega \text{: } \exists \, e \in \mathcal{J} \text{ with } se = e \text{ and } \chi(e) = 1 \right\}\end{equation}
    is closed in $\left\{ \chi \in \Omega \text{: } \chi(s^{-1}s) = 1 \right\}$ . The latter statement is equivalent to compactness of the set in (4.1) because $\left\{ \chi \in \Omega \text{: } \chi(s^{-1}s) = 1 \right\}$ is compact. This in turn is true if and only if there exists a finite subset $\left\{ e_1, \dotsc, e_n \right\} \subseteq \mathcal{J}$ with $se_i = e_i$ for all $1 \leq i \leq n$ and
    (4.2) \begin{equation} \left\{ \chi \in \Omega \text{: } \exists \, e \in \mathcal{J} \text{ with } se = e \text{ and } \chi(e) = 1 \right\} = \bigcup_{i=1}^n \Omega(e_i).\end{equation}
    We claim that (4.2) is equivalent to $\left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\} = \bigcup_{i=1}^n e_i$ . As $s e_i = e_i$ , we always have $\left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\} \supseteq \bigcup_{i=1}^n e_i$ . Assume that $\left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\} \subseteq \bigcup_{i=1}^n e_i$ . Given $\chi \in \Omega$ together with $e \in \mathcal{J}$ such that $se = e$ and $\chi(e) = 1$ , we must have $e \subseteq \bigcup_{i=1}^n e_i$ . As $\chi$ lies in $\Omega$ , $\chi(e) = 1$ implies that there exists $1 \leq i \leq n$ with $\chi(e_i) = 1$ . Hence (4.2) holds. Conversely, suppose that (4.2) holds. Take $x \in \textrm{dom}\ (s)$ with $s(x) = x$ . Then $\chi_x$ lies in the set on the left-hand side of (4.2), hence there exists $1 \leq i \leq n$ with $\chi_x(e_i) = 1$ . The latter implies that $x \in e_i$ . This shows $\left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\} \subseteq \bigcup_{i=1}^n e_i$ , as desired.
  2. (ii) [Reference Exel and Pardo25, Theorem 3.15] implies that $I_l \,{\bar{\ltimes}}\, \Omega$ is Hausdorff if and only if for all $t \in \bar{I}_l$ , the subset

    (4.3) \begin{equation} \left\{ \chi \in \Omega \text{: } \exists \, \varepsilon \in \bar{\mathcal{J}} \text{ with } t \varepsilon = \varepsilon \text{ and } \chi(\varepsilon) = 1 \right\}\end{equation}
    is closed in $\left\{ \chi \in \Omega \text{: } \chi(t^{-1}t) = 1 \right\}$ . First, we claim that the latter is equivalent to the statement that for all $s \in I_l$ , the subset
    (4.4) \begin{equation} \left\{ \chi \in \Omega \text{: } \exists \, \varepsilon \in \bar{\mathcal{J}} \text{ with } s \varepsilon = \varepsilon \text{ and } \chi(\varepsilon) = 1 \right\}\end{equation}
    is closed in $\left\{ \chi \in \Omega \text{: } \chi(s^{-1}s) = 1 \right\}$ . Indeed, a general element $t \in \bar{I}_l$ is of the form $s \delta$ for some $\delta \in \bar{\mathcal{J}}$ with $\delta \leq s^{-1}s$ . Now it is straightforward to see that the set in (4.3) coincides with the intersection of the set in (4.4) and $\Omega(\delta)$ . If the set in (4.4) is closed in $\left\{ \chi \in \Omega \text{: } \chi(s^{-1}s) = 1 \right\}$ , then its intersection with $\Omega(\delta)$ must be closed in $\left\{ \chi \in \Omega \text{: } \chi(s^{-1}s) = 1 \right\} \cap \Omega(\delta) = \left\{ \chi \in \Omega \text{: } \chi(t^{-1}t) = 1 \right\}$ . This shows our claim. Now the rest of the proof is similar as for (i).

In combination with Lemma 2.19, the following is immediate.

Corollary 4.2. Assume that $\mathfrak{C}$ is finitely aligned. Then $I_l \ltimes \Omega \cong I_l \,{\bar{\ltimes}}\, \Omega$ is Hausdorff if and only if for all $c, d \in \mathfrak{C}$ with $\mathfrak{d}(c) = \mathfrak{d}(d)$ and $\mathfrak{t}(c) = \mathfrak{t}(d)$ , there exists a finite subset $\left\{ x_1, \dotsc, x_n \right\} \subseteq \mathfrak{C}$ with $ \left\{ x \in \mathfrak{C} \text{: } cx = dx \right\} = \bigcup_{i=1}^n x_i \mathfrak{C}$ .

Remark 4.3. Lemma 4.1 and Corollary 4.2 explain the results in [Reference Spielberg69, Section 7] that $I_l \,{\bar{\ltimes}}\, \Omega$ is Hausdorff if $\mathfrak{C}$ is finitely aligned and right cancellative, or if $\mathfrak{C}$ embeds into a groupoid. In the first case, the set $\left\{ x \in \mathfrak{C} \text{: } cx = dx \right\}$ is either empty or we have $c = d$ , which implies that $\left\{ x \in \mathfrak{C} \text{: } cx = dx \right\} = \mathfrak{C}$ . In the second case, the set $\left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\}$ is either empty or we have $s \in \bar{\mathcal{J}}$ , in which case $\left\{ x \in \textrm{dom}\ (s) \text{: } s(x) = x \right\}$ coincides with $s^{-1}s$ , where we view the latter as a subset of $\mathfrak{C}$ .

Let us now consider topological freeness and effectiveness. Recall that an étale groupoid $\mathcal{G}$ is called effective if the interior of its isotropy subgroupoid coincides with the unit space, i.e., $\textrm{Iso}(\mathcal{G})^{\circ} = \mathcal{G}^{(0)}$ . Following [Reference KwaŚniewski and Meyer37, Definition 2.20], we call an Étale groupoid $\mathcal{G}$ topologically free if for every open bisection $\gamma$ with $\gamma \subseteq \mathcal{G} \setminus \mathcal{G}^{(0)}$ , $ \left\{ x \in \mathcal{G}^{(0)} \text{: } \mathcal{G}_x^x \cap \gamma \neq \emptyset \right\}$ has empty interior, or equivalently, $ \left\{ x \in {{\textrm{s}}}(\gamma) \text{: } \gamma x \notin \mathcal{G}_x^x \right\}$ is dense in ${{\textrm{s}}}(\gamma)$ . By [Reference KwaŚniewski and Meyer37, Lemma 2.23], $\mathcal{G}$ is topologically free if $\mathcal{G}$ is effective, and the converse holds if $\mathcal{G}$ is Hausdorff. Topological freeness for groupoids is of interest because it implies the intersection properties for essential groupoid C*-algebras (see [Reference KwaŚniewski and Meyer37, Section 7.5] for more information).

Now we set $\mathfrak{C}^{*,0} \mathrel{:=} \left\{ u \in \mathfrak{C}^* \text{: } \mathfrak{t}(u) = \mathfrak{d}(u) \right\}$ , and set $\mathfrak{C}^{*,0} \ltimes \Omega \mathrel{:=} \left\{ [u,\chi] \in I_l \ltimes \Omega \text{: } u \in \mathfrak{C}^{*,0} \right\}$ .

Theorem 4.4. The following are equivalent:

  1. (i) $I_l \ltimes \Omega$ is topologically free;

  2. (ii) $\mathfrak{C}^{*,0} \ltimes \Omega$ is topologically free;

  3. (iii) For all $\mathfrak{v} \in \mathfrak{C}^0$ , $u \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ , $f_1, \dotsc, f_n \in \mathcal{J}$ with $f_i \lneq \mathfrak{v} \mathfrak{C}$ for all $1 \leq i \leq n$ , $u z \in z \mathfrak{C}^*$ for all $z \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i$ implies that there exists $x \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i$ with $ux = x$ .

Proof.

(i) $\Rightarrow$ (ii): $\mathfrak{C}^{*,0} \ltimes \Omega$ is an open subgroupoid of $I_l \ltimes \Omega$ . Thus, an open bisection $\gamma$ of $\mathfrak{C}^{*,0} \ltimes \Omega$ with $\gamma \subseteq (\mathfrak{C}^{*,0} \ltimes \Omega) \setminus \Omega$ is also an open bisection of $I_l \ltimes \Omega$ contained in $(I_l \ltimes \Omega) \setminus \Omega$ . Moreover, $\gamma x \notin (I_l \ltimes \Omega)_x^x$ implies that $\gamma x \notin (\mathfrak{C}^{*,0} \ltimes \Omega)_x^x$ . This shows that

\begin{equation*} \left\{ x \in {{\textrm{s}}}(\gamma) \text{: } \gamma x \notin (I_l \ltimes \Omega)_x^x \right\} \subseteq \left\{ x \in {{\textrm{s}}}(\gamma) \text{: } \gamma x \notin (\mathfrak{C}^{*,0} \ltimes \Omega)_x^x \right\}.\end{equation*}

Hence $\mathfrak{C}^{*,0} \ltimes \Omega$ is topologically free if $I_l \ltimes \Omega$ is topologically free.

(ii) $\Rightarrow$ (i): Assume that $I_l \ltimes \Omega$ is not topologically free. Then we can find $s \in I_l$ and an open set $U \subseteq \Omega(s^{-1}s)$ with $[s,U] \subseteq (I_l \ltimes \Omega) \setminus \Omega$ and $[s,U] \subseteq \textrm{Iso}(I_l \ltimes \Omega)$ . As $\left\{ \chi_x \text{: } x \in \mathfrak{C} \right\}$ is dense in $\Omega$ , there exists $x \in \mathfrak{C}$ with $\chi_x \in U$ . $s.\chi_x = \chi_x$ implies that $s(x) = xu$ for some $u \in \mathfrak{C}^{*,0}$ . As $[s,\chi_x] \neq \chi_x$ , we conclude that $u \notin \mathfrak{C}^0$ . Set $V \mathrel{:=} \Omega(x \mathfrak{C}) \cap U$ . V is not empty, so that $x^{-1}.V \neq \emptyset$ . It is easy to see that $[x,\Omega(\mathfrak{d}(x))]^{-1} [s,V] [x,\Omega(\mathfrak{d}(x))] = [u, x^{-1}.V]$ . Moreover, $[x,\Omega(\mathfrak{d}(x))]^{-1} [s,V] [x,\Omega(\mathfrak{d}(x))]$ is contained in $\textrm{Iso}(I_l \ltimes \Omega) \setminus \Omega$ because $[s,V] \subseteq \textrm{Iso}(I_l \ltimes \Omega) \setminus \Omega$ . This means that $\mathfrak{C}^{*,0} \ltimes \Omega$ is not topologically free.

(ii) $\Rightarrow$ (iii): Assume $u z \in z \mathfrak{C}^*$ for all $z \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i$ . Set $U \mathrel{:=} \Omega(\mathfrak{v} \mathfrak{C}; f_1, \dotsc, f_n)$ . Then $[u,U] \subseteq \textrm{Iso}(\mathfrak{C}^{*,0} \ltimes \Omega)$ . As $\mathfrak{C}^{*,0} \ltimes \Omega$ is topologically free, there exists $\chi \in U$ with $[u,\chi] = \chi$ , i.e., there exists $e \in \mathcal{J}$ with $\chi(e) = 1$ and $ue = e$ . $\chi(\mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i) = 1$ implies that $e \not\subseteq \bigcup_{i=1}^n f_i$ . Hence we can choose $x \in e \setminus \bigcup_{i=1}^n f_i$ , and we have $ux = x$ .

(iii) $\Rightarrow$ (ii): First we claim that (iii) is equivalent to the following stronger statement:

(iii’) For all $\mathfrak{v} \in \mathfrak{C}^0$ , $u \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ , $e, f_1, \dotsc, f_n \in \mathcal{J}$ with $e, f_1, \dotsc, f_n \leq \mathfrak{v} \mathfrak{C}$ and $\bigcup_{i=1}^n f_i \subsetneq e$ , $u z \in z \mathfrak{C}^*$ for all $z \in e \setminus \bigcup_{i=1}^n f_i$ implies that there exists $x \in e \setminus \bigcup_{i=1}^n f_i$ with $ux = x$ .

Indeed, to prove (iii) $\Rightarrow$ (iii’), take $y \in e \setminus \bigcup_{i=1}^n f_i$ and set $\mathfrak{v} \mathrel{:=} \mathfrak{d}(y)$ . By assumption, $uy \in y \mathfrak{C}^*$ , and hence we have $uy = y \tilde{u}$ for some $\tilde{u} \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ . Set $f'_i \mathrel{:=} y \mathfrak{C} \cap f_i$ . Then $\bigcup_{i=1}^n f'_i \subsetneq y \mathfrak{C}$ implies that $\bigcup_{i=1}^n y^{-1} f'_i \subsetneq \mathfrak{v} \mathfrak{C}$ . For every $\tilde{x} \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n y^{-1} f'_i$ , we have by assumption $y \tilde{u} \tilde{x} = u y \tilde{x} \in y \tilde{x} \mathfrak{C}^*$ and thus $\tilde{u} \tilde{x} \in \tilde{x} \mathfrak{C}^*$ . Hence (iii) implies that there exists $x \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n y^{-1} f'_i$ with $\tilde{u} x = x$ . Then $y x \in e \setminus \bigcup_{i=1}^n f_i$ and $u y x = y \tilde{u} x = y x$ , as desired.

Now assume that (iii’) holds. Let $u \in \mathfrak{C}^{*,0}$ , $U = \Omega(e; f_1, \dotsc, f_n)$ , and assume that $[u,U] \subseteq \textrm{Iso}$ $(\mathfrak{C}^{*,0} \ltimes \Omega)$ . Then we must have $u z \in z \mathfrak{C}^*$ for all $z \in e \setminus \bigcup_{i=1}^n f_i$ . Hence (iii’) implies that there exists $x \in e \setminus \bigcup_{i=1}^n f_i$ with $ux = x$ . Then $\chi_x \in U$ because $x \in e \setminus \bigcup_{i=1}^n f_i$ . Moreover, $ux = x$ implies that $[u,\chi_x] = \chi_x \in \Omega$ . Hence $\mathfrak{C}^{*,0} \ltimes \Omega$ is topologically free.

We now consider $I_l \,{\bar{\ltimes}}\, \Omega$ . As before, we set $\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega \mathrel{:=} \left\{ [u,\chi]^{\bar{\sim}} \in I_l \,{\bar{\ltimes}}\, \Omega \text{: } u \in \mathfrak{C}^{*,0} \right\}$ .

Theorem 4.5. The following are equivalent:

  1. (i) $I_l \,{\bar{\ltimes}}\, \Omega$ is effective;

  2. (ii) $\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega$ is effective;

  3. (iii) For all $\mathfrak{v} \in \mathfrak{C}^0$ , $u \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ , $f_1, \dotsc, f_n \in \mathcal{J}$ with $f_i \lneq \mathfrak{v} \mathfrak{C}$ for all $1 \leq i \leq n$ , $u z \in z \mathfrak{C}^*$ for all $z \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i$ implies that $ux = x$ for all $x \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i$ .

Proof.

  1. (i) $\Rightarrow$ (ii) is clear because $\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega$ is an open subgroupoid of $I_l \,{\bar{\ltimes}}\, \Omega$ .

  2. (ii) $\Rightarrow$ (i): Suppose that $I_l \,{\bar{\ltimes}}\, \Omega$ is not effective. Then there exist $s \in I_l$ , $U \mathrel{:=} \Omega(e; f_1, \dotsc, f_n)$ and $\chi \in U$ with $[s,U] \subseteq \textrm{Iso}(I_l \,{\bar{\ltimes}}\, \Omega)$ and $[s,\chi] \neq \chi$ . Set $\varepsilon \mathrel{:=} e \setminus \bigcup_{i=1}^n f_i$ . $[s,\chi] \neq \chi$ implies that $s \varepsilon \neq \varepsilon$ , i.e., there exists $x \in \varepsilon$ with $s(x) \neq x$ . We have $\chi_x \in U$ , and $s(x) \neq x$ implies $[s,\chi_x] \neq \chi_x$ . However, $s.\chi_x = \chi_x$ , and thus $s(x) = xu$ for some $u \in \mathfrak{C}^{*,0}$ with $\mathfrak{t}(u) = \mathfrak{d}(u) = \mathfrak{d}(x)$ . We deduce $x \neq xu$ , i.e., $u \neq \mathfrak{d}(x)$ . Set $V \mathrel{:=} \Omega(x) \cap U$ . Then $\chi_x \in V$ , so V is not empty. Moreover, $[u,c^{-1}.V] = [x,\Omega(\mathfrak{d}(x))]^{-1} [s,V] [x,\Omega(\mathfrak{d}(x))]$ is contained in $\textrm{Iso}(\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega)$ . We have $[u,\chi_{\mathfrak{d}(x)}] = [x,\Omega(\mathfrak{d}(x))]^{-1} [s,\chi_x] [x,\Omega(\mathfrak{d}(x))] \in [u,c^{-1}.V]$ and $[u,\chi_{\mathfrak{d}(x)}] \neq \chi_{\mathfrak{d}(x)}$ because $u \mathfrak{d}(x) \neq \mathfrak{d}(x)$ . It follows that $\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega$ is not effective.

To prove (ii) $\Leftrightarrow$ (iii), we first show that (iii) is equivalent to the following stronger statement:

  1. (iii’) For all $\mathfrak{v} \in \mathfrak{C}^0$ , $u \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ , $e, f_1, \dotsc, f_n \in \mathcal{J}$ with $e, f_1, \dotsc, f_n \leq \mathfrak{v} \mathfrak{C}$ and $\bigcup_{i=1}^n f_i \subsetneq e$ , $u z \in z \mathfrak{C}^*$ for all $z \in e \setminus \bigcup_{i=1}^n f_i$ implies that $ux = x$ for all $x \in e \setminus \bigcup_{i=1}^n f_i$ .

Indeed, to prove (iii) $\Rightarrow$ (iii’), take $x \in e \setminus \bigcup_{i=1}^n f_i$ and set $\mathfrak{v} \mathrel{:=} \mathfrak{d}(x)$ . By assumption, $ux \in x \mathfrak{C}^*$ , and hence we have $ux = x \tilde{u}$ for some $\tilde{u} \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ . Set $f'_i \mathrel{:=} x \mathfrak{C} \cap f_i$ . Then $\bigcup_{i=1}^n f'_i \subsetneq x \mathfrak{C}$ implies that $\bigcup_{i=1}^n x^{-1} f'_i \subsetneq \mathfrak{v} \mathfrak{C}$ . For every $\tilde{x} \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n x^{-1} f'_i$ , we have by assumption $x \tilde{u} \tilde{x} = u x \tilde{x} \in x \tilde{x} \mathfrak{C}^*$ and thus $\tilde{u} \tilde{x} \in \tilde{x} \mathfrak{C}^*$ . Hence (iii) implies $\tilde{u} = \tilde{u} \mathfrak{v} = \mathfrak{v}$ and thus $u x = x \tilde{u} = x$ . As x was an arbitrary element of $e \setminus \bigcup_{i=1}^n f_i$ , we are done.

Now let us prove (ii) $\Leftrightarrow$ (iii). $\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega$ is effective if and only if for all $u \in \mathfrak{C}^{*,0}$ and $\varepsilon = e \setminus \bigcup_{i=1}^n f_i \in \bar{\mathcal{J}}$ , $[u,\Omega(\varepsilon)] \subseteq \textrm{Iso}(\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega)$ implies $[u,\Omega(\varepsilon)] = \Omega(\varepsilon)$ . $[u,\Omega(\varepsilon)] \subseteq \textrm{Iso}(\mathfrak{C}^{*,0} \,{\bar{\ltimes}}\, \Omega)$ holds if and only if $uz \in z \mathfrak{C}^*$ for all $z \in \varepsilon$ , whereas $[u,\Omega(\varepsilon)] = \Omega(\varepsilon)$ holds if and only if $u \varepsilon = \varepsilon$ , i.e., $ux = x$ for all $x \in \varepsilon$ . We conclude that (ii) and (iii’) are equivalent.

The following are immediate consequences.

Corollary 4.6. If $I_l \,{\bar{\ltimes}}\, \Omega$ is effective, then $I_l \ltimes \Omega$ is topologically free.

Corollary 4.7. Assume that $\mathfrak{C}$ is finitely aligned.

  1. (i) $I_l \ltimes \Omega \cong I_l \,{\bar{\ltimes}}\, \Omega$ is topologically free if and only if for all $\mathfrak{v} \in \mathfrak{C}^0$ , $u \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ , $c_1, \dotsc, c_n \in \mathfrak{v} \mathfrak{C} \setminus \mathfrak{v} \mathfrak{C}^*$ , $u z \in z \mathfrak{C}^*$ for all $z \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n c_i \mathfrak{C}$ implies that there exists $x \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n c_i \mathfrak{C}$ with $ux = x$ .

  2. (ii) $I_l \ltimes \Omega \cong I_l \,{\bar{\ltimes}}\, \Omega$ is effective if and only if for all $\mathfrak{v} \in \mathfrak{C}^0$ , $u \in \mathfrak{v} \mathfrak{C}^* \mathfrak{v}$ , $c_1, \dotsc, c_n \in \mathfrak{v} \mathfrak{C} \setminus \mathfrak{v} \mathfrak{C}^*$ , $u z \in z \mathfrak{C}^*$ for all $z \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n c_i \mathfrak{C}$ implies that $ux = x$ for all $x \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n c_i \mathfrak{C}$ .

We also note the following special case, where our conditions simplify.

Corollary 4.8. Assume that for all $\mathfrak{v} \in \mathfrak{C}^0$ , there exist $f_1, \dotsc, f_n \in \mathcal{J}$ with $\mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i = \mathfrak{v} \mathfrak{C}^*$ . Then the following are equivalent:

  1. (i) $I_l \,{\bar{\ltimes}}\, \Omega$ is effective.

  2. (ii) $I_l \ltimes \Omega$ is topologically free.

  3. (iii) $\mathfrak{C}^{*,0} = \mathfrak{C}^0$ .

Proof. (i) $\Rightarrow$ (ii) has been noted above. Let us prove (ii) $\Rightarrow$ (iii). We have for all $z \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i = \mathfrak{v} \mathfrak{C}^*$ that $uz = z (z^{-1} u z) \in z \mathfrak{C}^*$ . Hence, Theorem 4.4 (iii) implies that there exists $x \in \mathfrak{v} \mathfrak{C} \setminus \bigcup_{i=1}^n f_i = \mathfrak{v} \mathfrak{C}^*$ with $ux = x$ . Hence, $u = ux x^{-1} = x x^{-1} = \mathfrak{v}$ . (iii) $\Rightarrow$ (i) is immediate from Theorem 4.5.

Remark 4.9. Theorems 4.4 and 4.5 generalise [Reference Laca and Sehnem40, Theorem 5.9].

In combination with [Reference KwaŚniewski and Meyer37, Theorem 7.29] (and the explanations following Theorem 7.29 in [Reference KwaŚniewski and Meyer37]), the following are consequences of our results above.

Corollary 4.10. If the conditions in Lemma 4.1 (i) and Theorem 4.4 (iii) are satisfied, then $C^*_r(I_l \ltimes \Omega)$ has the intersection property.

If the conditions in Lemma 4.1 (ii) and Theorem 4.5 (iii) are satisfied, then $C^*_r(I_l \,{\bar{\ltimes}}\, \Omega)$ has the intersection property.

Suppose that $\mathfrak{C}$ is finitely aligned. If the condition in Corollary 4.2 and one of the conditions in Corollary 4.7 are satisfied, then $C^*_r(I_l \ltimes \Omega) \cong C^*_r(I_l \,{\bar{\ltimes}}\, \Omega)$ has the intersection property.

Remark 4.11. It is also possible to give a characterisation for minimality of $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ by formulating a characterisation when $\Omega = \partial \Omega$ along the lines of [Reference Cuntz, Echterhoff, Li and Yu15, Lemma 5.7.19] and then applying our characterisation for minimality of $I_l \ltimes \partial \Omega$ and $I_l \,{\bar{\ltimes}}\, \partial \Omega$ (see Lemma 5.4).

Remark 4.12. It would also be possible to formulate sufficient criteria for local contractiveness of $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ . However, this happens only in rather special situations (see Proposition 6.32 and Corollary 6.33, for example). For instance, in the setting of Corollary 4.8, $I_l \ltimes \Omega$ and $I_l \,{\bar{\ltimes}}\, \Omega$ and are never locally contractive because the assumptions in Corollary 4.8 imply that $\left\{ \chi_{\mathfrak{v}} \right\}$ is open for all $\mathfrak{v} \in \mathfrak{C}^0$ .

5. Properties of the boundary groupoid

We characterise when $I_l \ltimes \partial \Omega$ and $I_l \,{\bar{\ltimes}}\, \partial \Omega$ are Hausdorff or minimal, when $I_l \ltimes \partial \Omega$ is topologically free, when $I_l \,{\bar{\ltimes}}\, \partial \Omega$ is effective, and we give a sufficient condition for local contractiveness of $I_l \ltimes \partial \Omega$ and $I_l \,{\bar{\ltimes}}\, \partial \Omega$ . These properties have consequences for the boundary quotients (see Corollary 5.20).

Note that if $I_l$ does not contain zero, then $\# \mathfrak{C}^0 = 1$ and $\partial \Omega$ degenerates to a point. Because of this, it suffices in the following to focus on the case when $I_l$ contains zero.

We first consider the Hausdorff property. The following is an application of [Reference Exel and Pardo25, Theorem 3.16] because $I_l \ltimes \partial \Omega$ is the tight groupoid of the inverse semigroup $I_l$ .

Lemma 5.1. $I_l \ltimes \partial \Omega$ is Hausdorff if and only if for all $s \in I_l$ there exist $e_1, \dotsc, e_n \in \mathcal{J}$ with $s e_i = e_i$ such that for all $0 \neq e \in \mathcal{J}$ with $se = e$ , there exists $1 \leq i \leq n$ with $e e_i \neq 0$ .

Now we characterise when $I_l \,{\bar{\ltimes}}\, \partial \Omega$ is Hausdorff.

Lemma 5.2. $I_l \,{\bar{\ltimes}}\, \partial \Omega$ is Hausdorff if and only if for all $s \in I_l$ there exist $\varepsilon_1, \dotsc, \varepsilon_n \in \bar{\mathcal{J}}$ with $s \varepsilon_i = \varepsilon_i$ such that for all $0 \neq e \in \mathcal{J}$ with $s e = e$ , there exists $1 \leq i \leq n$ such that $e \varepsilon_i \neq 0$ .

Proof. We make use of the identification $I_l \,{\bar{\ltimes}}\, \partial \Omega \cong \bar{I}_l \ltimes \partial \Omega$ (see Remark 2.16). [Reference Exel and Pardo25, Theorem 3.15], applied to $\bar{I}_l \curvearrowright \partial \Omega$ , implies that $I_l \,{\bar{\ltimes}}\, \partial \Omega$ is Hausdorff if and only if for all $s \in \bar{I}_l$ there exist $\varepsilon_1, \dotsc, \varepsilon_n \in \bar{\mathcal{J}}$ with $s \varepsilon_i = \varepsilon_i$ such that for all $\chi \in \partial \Omega$ , $\varepsilon \in \bar{\mathcal{J}}$ with $\chi(\varepsilon) = 1$ and $s \varepsilon = \varepsilon$ , there exists $1 \leq i \leq n$ such that $\chi(\varepsilon_i) = 1$ . We may assume that $s \in I_l$ in this statement because every $\bar{s} \in \bar{I}_l$ is of the form $s \delta$ for some $s \in I_l$ and $\delta \in \bar{\mathcal{J}}$ , and we can form products of $\varepsilon$ and $\varepsilon_i$ with $\delta$ . Next, we claim that the statement is equivalent to the following: For all $s \in I_l$ , there exist $\varepsilon_1, \dotsc, \varepsilon_n \in \bar{\mathcal{J}}$ with $s \varepsilon_i = \varepsilon_i$ such that for all $\chi \in \Omega_{\max}$ , $\varepsilon \in \bar{\mathcal{J}}$ with $\chi(\varepsilon) = 1$ and $s \varepsilon = \varepsilon$ , there exists $1 \leq i \leq n$ such that $\chi(\varepsilon_i) = 1$ . Indeed, given $\chi \in \partial \Omega$ , we can always find $\eta_{\lambda} \in \Omega_{\max}$ with $\chi = \lim_{\lambda} \eta_{\lambda}$ . We may then assume that $\eta_{\lambda}(\varepsilon) = 1$ for all $\lambda$ , and then deduce that for all $\lambda$ , there exists $1 \leq i \leq n$ with $\eta_{\lambda}(\varepsilon_i) = 1$ . By passing to a subnet if necessary, we arrange that there exists $1 \leq i \leq n$ with $\eta_{\lambda}(\varepsilon_i) = 1$ for all $\lambda$ , and thus $\chi(\varepsilon_i) = 1$ . Now we claim that our new statement is equivalent to the following: For all $s \in I_l$ , there exist $\varepsilon_1, \dotsc, \varepsilon_n \in \bar{\mathcal{J}}$ with $s \varepsilon_i = \varepsilon_i$ such that for all $\chi \in \Omega_{\max}$ , $0 \neq e \in \mathcal{J}$ with $\chi(e) = 1$ and $s e = e$ , there exists $1 \leq i \leq n$ such that $\chi(\varepsilon_i) = 1$ . Indeed, given $\chi \in \Omega_{\max}$ and $\varepsilon \in \bar{\mathcal{J}}$ with $\chi(\varepsilon) = 1$ , Lemma 2.21 implies that there exists $e \in \mathcal{J}$ with $\chi(e) = 1$ and $e \leq \varepsilon$ . Finally, we claim that our statement is equivalent to the desired one: For all $s \in I_l$ , there exist $\varepsilon_1, \dotsc, \varepsilon_n \in \bar{\mathcal{J}}$ with $s \varepsilon_i = \varepsilon_i$ such that for all $0 \neq e \in \mathcal{J}$ with $s e = e$ , there exists $1 \leq i \leq n$ such that $e \varepsilon_i \neq 0$ . To see “ $\Rightarrow$ ”, if there exists $0 \neq e \in \mathcal{J}$ with $e \varepsilon_i = 0$ for all i, then Lemma 2.21 yields a character $\chi \in \Omega_{\max}$ with $\chi(e) = 1$ , and we obtain $\chi(\varepsilon_i) = 0$ for all i. For “ $\Leftarrow$ ”, assume that there exist $\chi \in \Omega_{\max}$ , $0 \neq e \in \mathcal{J}$ with $\chi(e) = 1$ and $s e = e$ such that