Hausdorff Groupoid Research Articles

On the Construction of a Groupoid from an Ample Hausdorff Groupoid with Twisted Steinberg Algebra not Isomorphic to its Non-twisted Steinberg Algebra

This study introduces an ample Hausdorff groupoid $\hat{A} \rtimes \mathcal{R}$ extracted from an ample Hausdorff groupoid $\mathcal{G}$ and a unital commutative ring $R$; a Hausdorff groupoid $D$ which is the discrete twist over $\hat{A} \rtimes \mathcal{R}$. In the groupoid C*-algebra perspective, when $R = \mathbb{C}$ there is an isomorphism between the non-twisted groupoid C*-algebra $(C^*(\mathcal{G}))$ and the twisted groupoid C*-algebra $(C^*(\hat{A} \rtimes \mathcal{R};D))$. However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra $(A_R(\mathcal{G}))$ and the twisted Steinberg algebra $(A_R(D; \hat{A} \rtimes \mathcal{R}))$ are non-isomorphic.

Representing topological full groups in Steinberg algebras and C*-algebras

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group.

The local bisection hypothesis for twisted groupoid C*-algebras

In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid G being effective. One of these conditions is that G satisfies the C*-algebraic local bisection hypothesis; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras.

Zak transform for groupoids with abelian isotropy

Let 𝒢 be a second countable locally compact Hausdorff groupoid with abelian isotropy groups and 𝓛 be a lattice bundle in the isotropy subgroupoid 𝒢 ' of 𝒢. In this paper, we define a Zak transform on (...)

Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring $R$. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid $G$ and each locally constant $2$-cocycle $\sigma$ on $G$ taking values in the units $R^\times$, we study the algebra $A_R(G,\sigma)$ consisting of locally constant compactly supported $R$-valued functions on $G$, with convolution and involution "twisted" by $\sigma$. We also introduce a "discretised" analogue of a twist $\Sigma$ over a Hausdorff \'etale groupoid $G$, and we show that there is a one-to-one correspondence between locally constant $2$-cocycles on $G$ and discrete twists over $G$ admitting a continuous global section. Given a discrete twist $\Sigma$ arising from a locally constant $2$-cocycle $\sigma$ on an ample Hausdorff groupoid $G$, we construct an associated twisted Steinberg algebra $A_R(G;\Sigma)$, and we show that it coincides with $A_R(G,\sigma^{-1})$. Given any discrete field $\mathbb{F}_d$, we prove a graded uniqueness theorem for $A_{\mathbb{F}_d}(G,\sigma)$, and under the additional hypothesis that $G$ is effective, we prove a Cuntz--Krieger uniqueness theorem and show that simplicity of $A_{\mathbb{F}_d}(G,\sigma)$ is equivalent to minimality of $G$.

An algebraic characterisation of ample type I groupoids

We give algebraic characterisations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of its Boolean inverse semigroup of compact open local bisections. It yields in turn algebraic characterisations of both properties for inverse semigroups with meets in terms of subquotients of their Booleanisation.

Reconstruction of Twisted Steinberg Algebras

Abstract We show how to recover a discrete twist over an ample Hausdorff groupoid from a pair consisting of an algebra and what we call a quasi-Cartan subalgebra. We identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), and we prove that the assignment of twisted Steinberg algebras to such twists and our construction of a twist from a quasi-Cartan pair are mutually inverse. We identify the algebraic pairs that correspond to effective groupoids and to principal groupoids. We also indicate the scope of our results by identifying large classes of twists for which the local bisection hypothesis holds automatically.

Localised Module Frames and Wannier Bases from Groupoid Morita Equivalences

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is étale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schrödinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.

Non-simple Purely Infinite Steinberg Algebras with Applications to Kumjian–Pask Algebras

We characterize properly purely infinite Steinberg algebras \(A_K({\mathcal {G}})\) for strongly effective, ample Hausdorff groupoids \({\mathcal {G}}\). As an application, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian–Pask algebra \(\mathrm {KP}_K(\Lambda )\) in case \(\Lambda \) is a strongly aperiodic k-graph. In particular, for unital cases, we give simple graph-theoretic criteria for the (proper) pure infiniteness of \(\mathrm {KP}_K(\Lambda )\). Furthermore, since the complex Steinberg algebra \(A_{\mathbb {C}}({\mathcal {G}})\) is a dense subalgebra of the reduced groupoid \(C^*\)-algebra \(C^*_r({\mathcal {G}})\), we focus on the problem that “when does the proper pure infiniteness of \(A_{\mathbb {C}}({\mathcal {G}})\) imply that of \(C^*_r({\mathcal {G}})\) in the \(C^*\)-sense?”. In particular, we show that if the Kumjian–Pask algebra \(\mathrm {KP}_{\mathbb {C}}(\Lambda )\) is purely infinite, then so is \(C^*(\Lambda )\) in the sense of Kirchberg–Rørdam.

Essential crossed products for inverse semigroup actions: simplicity and pure infiniteness

We study simplicity and pure infiniteness criteria for \mathrm{C}^* -algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over étale not necessarily Hausdorff groupoids. Inspired by recent work of R. Exel and D. R. Pitts ["Characterizing groupoid \mathrm{C}^* -algebras of non-Hausdorff étale groupoids", Preprint (2019); [arXiv: 1901.09683](arXiv: 1901.09683)], we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced \mathrm{C}^* -algebras coincide. Our most general simplicity and pure infiniteness criteria apply to aperiodic \mathrm{C}^* -inclusions equipped with supportive generalised expectations. We thoroughly discuss the relationship between aperiodicity, detection of ideals, purely outer inverse semigroup actions, and non-triviality conditions for dual groupoids.

On the irreducibility of induced representations of some groupoid $$C^*$$-algebras

Let $${\mathcal {G}}$$ be a topological locally compact Hausdorff and second countable groupoid with a Haar system and $${\mathcal {K}}$$ a proper subgroupoid of $${\mathcal {G}}$$ with a Haar system too. $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair if for any u in the unit space, the pair of isotropy groups $$({\mathcal {G}}(u), {\mathcal {K}}(u))$$ is a Gelfand pair. In this work, we prove, when $$({\mathcal {G}},{\mathcal {K}})$$ is an internally Gelfand pair that any representation of $${\mathcal {C}}^{*} ({\mathcal {G}}, {\mathcal {K}},\lambda , \alpha )$$ induced from a positive definite spherical function on an isotropy group is irreducible. We introduce also the notion of spherical bundle and after identifying it with the spectrum of some $$C^*$$ -algebra, we obtain some properties of its topology.

Equivariant $KK$-theory of $r$-discrete groupoids and inverse semigroups

For an r-discrete Hausdorff groupoid 𝒢 and an inverse semigroup S of slices of 𝒢 there is an isomorphism between 𝒢-equivariant KK-theory and compatible S-equivariant KK-theory. We use it to define descent homomorphisms for S, and indicate a Baum–Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in KK-theory.

A Going-Down principle for ample groupoids and the Baum-Connes conjecture

We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory. Moreover, we provide an application to the Baum-Connes conjecture for ample groupoids which are strongly amenable at infinity. This result in turn is then used to relate the Baum-Connes conjecture for an ample groupoid group bundle which is strongly amenable at infinity to the Baum-Connes conjecture for the fibres.

Stone duality and quasi‐orbit spaces for generalised C∗‐inclusions

Let $A$ and $B$ be $C^*$-algebras with $A\subseteq M(B)$. Exploiting the duality between sober spaces and spatial locales, and the adjunction between restriction and induction for ideals in $A$ and $B$, we identify conditions that allow to define a quasi-orbit space and a quasi-orbit map for $A\subseteq M(B)$. These objects generalise classical notions for group actions. We characterise when the quasi-orbit space is an open quotient of the primitive ideal space of $A$ and when the quasi-orbit map is open and surjective. We apply these results to cross section $C^*$-algebras of Fell bundles over locally compact groups, regular $C^*$-inclusions, tensor products, relative Cuntz--Pimsner algebras, and crossed products for actions of locally compact Hausdorff groupoids and quantum groups.

Equivariant [formula omitted]-theory for non-Hausdorff groupoids

We give a detailed and unified survey of equivariant KK-theory over locally compact, second countable, locally Hausdorff groupoids. We indicate precisely how the “classical” proofs relating to the Kasparov product can be used almost word-for-word in this setting, and give proofs for several results which do not currently appear in the literature.

Finiteness spaces, étale groupoids and their convolution algebras

Given a ring R, we extend Ehrhard’s linearization process by associating to any pre-finiteness space an R-module endowed with a Lefschetz topology. For a semigroup in the category of pre-finiteness spaces, one can endow this R-module with the convolution product to obtain an R-algebra. As examples of pre-finiteness spaces, we study topological spaces with bounded subsets (i.e., included in a compact) taken to be the finitary subsets. We prove that we obtain a finiteness space from any hemicompact space via this construction. As a corollary, any etale Hausdorff groupoid induces a semigroup in pre-finiteness spaces and its associated convolution algebra is complete in the hemicompact case. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.

The orbit spaces of groupoids whose C⁎-algebras are CCR

Let G be a second-countable locally compact Hausdorff groupoid with a continuous Haar system. We show that if the groupoid C⁎-algebra of G is CCR then the orbits of G are closed and the stabilizers of G are CCR. In particular, we remove the assumption of amenability in a theorem of Clark.

Simple flat Leavitt path algebras are von Neumann regular

For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40 years ago who called such rings SF-rings (i.e. simple modules are flat). In this note we show that an SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids, this implies that the graphs are acyclic. Combining with the Abrams–Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi’s question in positive for the class of Leavitt path algebras.

Harmonic Analysis on Internally Gelfand Pairs Associated to Groupoids

Let G be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and K a proper subgroupoid of G with a Haar system too. (G, K) is an internally Gelfand pair if for any u in the unit space, the algebra of bi-K(u)-invariant functions on G(u) is commutative under convolution. In this work, we give some characterizations of these pairs and extend to this context some classical results of harmonic analysis.

Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed subspaces of their unit spaces are all effective. For such a groupoid, we completely describe the ideal lattice of the associated Steinberg algebra over any commutative ring with identity. Our results are new even for the special case of Leavitt path algebras; so we describe explicitly what they say in this context, and give two concrete examples.