A uniqueness theorem for twisted groupoid C*-algebras
Becky Armstrong
备注：
26 pages. This version matches the version in the Journal of Functional Analysis. The author would like to thank the anonymous referee for their careful reading and helpful suggestions
We present a uniqueness theorem for the reduced C*-algebra of a twist $\mathcal{E}$ over a Hausdorff étale groupoid $\mathcal{G}$. We show that the interior $\mathcal{I}^\mathcal{E}$ of the isotropy of $\mathcal{E}$ is a twist over the interior $\mathcal{I}^\mathcal{G}$ of the isotropy of $\mathcal{G}$, and that the reduced twisted groupoid C*-algebra $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ embeds in $C_r^*(\mathcal{G}; \mathcal{E})$. We also investigate the full and reduced twisted C*-algebras of the isotropy groups of $\mathcal{G}$, and we provide a sufficient condition under which states of (not necessarily unital) C*-algebras have unique state extensions. We use these results to prove our uniqueness theorem, which states that a C*-homomorphism of $C_r^*(\mathcal{G}; \mathcal{E})$ is injective if and only if its restriction to $C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ is injective. We also show that if $\mathcal{G}$ is effective, then $C_r^*(\mathcal{G}; \mathcal{E})$ is simple if and only if $\mathcal{G}$ is minimal.
The second author showed how Katsura's construction of the C*-algebra of a topological graph E may be twisted by a Hermitian line bundle L over the edge space E. The correspondence defining the algebra is obtained as the completion of the compactly supported continuous sections of L. We prove that the resulting C*-algebra is isomorphic to a twisted groupoid C*-algebra where the underlying groupoid is the Renault-Deaconu groupoid of the topological graph with Yeend's boundary path space as its unit space.
This article extends the main results of the publication arXiv:2001.01312 to the case of a twisted groupoid. More precisely, it gives a decomposition of the C*-algebra of a twisted locally compact groupoid with Haar system in presence of a normal subgroupoid. When the normal subgroupoid and the twist over it are abelian, one obtains another twisted groupoid C*-algebra. This is applied to C*-algebraic deformation quantization and to multiplier representations of locally compact abelian groups.
Given a saturated Fell bundle A over an inverse semigroup S which is semi-abelian in the sense that the fibers over the idempotents of S are commutative, we construct a twisted etale groupoid L such that A can be recovered from L in a canonical way. As an application we recover most of Renault's recent result on the classification of Cartan subalgebras of C*-algebras through twisted etale groupoids.
The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.
Let $C^*$-algebra that is acted upon by a compact abelian group. We show that if the fixed-point algebra of the action contains a Cartan subalgebra $D$ satisfying an appropriate regularity condition, then $A$ is the reduced $C^*$-algebra of a groupoid twist. We further show that the embedding $D \hookrightarrow A$ is uniquely determined by the twist. These results generalize Renault's results on Cartan subalgebras of $C^*$-algebras.
Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, Σ)$ for any twist $Σ$ over $G$. However, there are many groupoids which are not topologically principal, yet their (twisted) $C^*$-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle $c$ on $G$ and a subgroupoid $S \subseteq G$ under which $C^*_r(S, c)$ is Cartan in $C^*_r(G, c)$. When $G$ is a discrete group, we also describe the Weyl groupoid and twist associate…
We investigate recent uniqueness theorems for reduced $C^*$-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full $C^*$-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.
We introduce the Haagerup property for twisted groupoid $C^*$-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of `the Haagerup trick' we prove that this property is equivalent to the Haagerup property of the reduced crossed product $C^*$-algebra with respect to the canonical conditional expectation $E$. This extends a theorem of Dong and Ruan for discrete group actions, and implies that a given Cartan inclusion of separable $C^*$-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup…
We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit natural Hilbert space representations. We show the relevance of the orbit closure structure of the unit space of the groupoid in dealing with spectra, norms, numerical ranges and $ε$-pseudospectra of the resulting operators. As an example, we treat a class of pseudo-differential operators introduced recently, associated to group actions. We also prove a Deco…
The goal of this paper is to construct a calculus whose higher indices are naturally elements in the twisted K-theory groups for Lie groupoids. Given a Lie groupoid $G$ and a $PU(H)$-valued groupoid cocycle, we construct an algebra of projective pseudodifferential operators. The subalgebra of regularizing operators identifies with the naturally associated smooth convolution algebra of the associated twisted gerbe. We develop the associated symbolic calculus, symbol short exact sequences and existence of parametrices. In particular the algebra of projective operators appears as a quantization o…
We develop equivariant KK-theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K-theory.
Given a normal subgroup bundle $\mathcal A$ of the isotropy bundle of a groupoid $Σ$, we obtain a twisted action of the quotient groupoid $Σ/\mathcal A$ on the bundle of group $C^*$-algebras determined by $\mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Σ)$. Restricting to the case where $\mathcal A$ is abelian, we describe $C^*(Σ)$ as the $C^*$-algebra associated to a $\mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Σ/\mathcal A$ on the Pontryagin dual space of $\mathcal A$. We give some illustrative examples of thi…
We introduce a new notion of twisted actions of inverse semigroups and show that they correspond bijectively to certain regular Fell bundles over inverse semigroups, yielding in this way a structure classification of such bundles. These include as special cases all the stable Fell bundles. Our definition of twisted actions properly generalizes a previous one introduced by Sieben and corresponds to Busby-Smith twisted actions in the group case. As an application we describe twisted etale groupoid C*-algebras in terms of crossed products by twisted actions of inverse semigroups and show that Sie…
In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by the full uniqueness property of its groupoid C*-algebra.
We consider fiberwise singly generated Fell-bundles over etale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra of the Fell bundle. We study the Kubo-Martin-Schwinger equilibrium states for this dynamics. Following work of Neshveyev on equilibrium states on groupoid C*-algebras, we describe the equilibrium states of the cross-sectional algebra in terms of measurable fields of traces on the C*-algebras of the restrictions of the Fell bundle to the isotropy subgroups of the groupoid. As a special case, we obtain a des…
For a second-countable locally compact Hausdorff étale groupoid $\mathcal{G}$ with a continuous $2$-cocycle $σ$ we find conditions that guarantee that $\ell^1 (\mathcal{G},σ)$ has a unique $C^*$-norm.
We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary…
Inspired by recent papers on twisted $K$-theory, we consider in this article the question of when a twist $\mathcal{R}$ over a locally compact Hausdorff groupoid $\mathcal{G}$ (with unit space a CW-complex) admits a twisted vector bundle, and we relate this question to the Brauer group of $\mathcal{G}$. We show that the twists which admit twisted vector bundles give rise to a subgroup of the Brauer group of $\mathcal{G}$. When $\mathcal{G}$ is an étale groupoid, we establish conditions (involving the classifying space $B\mathcal{G}$ of $\mathcal{G}$) which imply that a torsion twist $\mathcal{…
We show that every groupoid C*-algebra is isomorphic to its opposite, and deduce that there exist C*-algebras that are not stably isomorphic to groupoid C*-algebras, though many of them are stably isomorphic to twisted groupoid C*-algebras. We also prove that the opposite algebra of a section algebra of a Fell-bundle over a groupoid is isomorphic to the section algebra of a natural opposite bundle.
We prove a uniqueness theorem and give a characterization of simplicity for Steinberg algebras associated to non-Hausdorff ample groupoids. We also prove a uniqueness theorem and give a characterization of simplicity for the C*-algebra associated to non-Hausdorff étale groupoids. Then we show how our results apply in the setting of tight representations of inverse semigroups, groups acting on graphs, and self-similar actions. In particular, we show that C*-algebra and the complex Steinberg algebra of the self-similar action of the Grigorchuk group are simple but the Steinberg algebra with coef…
Given a not-necessarily Hausdorff, topologically free, twisted étale groupoid $(G, L)$, we consider its "essential groupoid C*-algebra", denoted $C^*_{ess}(G, L)$, obtained by completing $C_c(G, L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0), C^*_{ess}(G, L))$ is then proven to satisfy a list of properties characterizing it as what we call a "weak Cartan inclusion". We then prove that every weak Cartan inclusion $(A, B)$, with $B$ separable, is modeled by a topologically free, twisted étale groupoid, as above…
This paper investigates the $\mathrm{K}$-theory of twisted groupoid $\mathrm{C}^*$-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to an isomorphism between the $\mathrm{K}$-theory groups of the respective twisted groupoid $\mathrm{C}^*$-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault-Deaconu groupoids and $P$-graph algebras.
Let E be a second-countable, locally compact, Hausdorff groupoid equipped with an action of T such that G:=E/T is a principal groupoid with Haar system λ. The twisted groupoid C*-algebra C*(E;G,λ) is a quotient of the C*-algebra of E obtained by completing the space of T-equivariant functions on E. We show that C*(E;G,λ) is postliminal if and only if the orbit space of G is T_0 and that C*(E;G, λ) is liminal if and only if the orbit space is T_1. We also show that C*(E;G, λ) has bounded trace if and only if G is integrable and that C*(E;G, λ) is a Fell algebra if and only if G is Cartan. Let \…
We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geo…
We consider the boundary-path groupoids of topological higher-rank graphs. We show that the all such groupoids are topologically amenable. We deduce that the C*-algebras of topological higher-rank graphs are nuclear and prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem. We then provide a necessary and sufficient condition for simplicity of a topological higher-rank graph C*-algebra, and a condition under which it is also purely infinite.
Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra and then, when $\mathcal{G}$ is graded by a cocycle, we study graded ideals in $A_R(\mathcal {G})$. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.
An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a lab…
We examine a semigroup analogue of the Kumjian-Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we show how to represent semigroups with distinguished normal subsemigroups as `slice-sections' of groupoid bundles.
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 tw…
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 $σ$ on $G$ taking values in the units $R^\times$, we study the algebra $A_R(G,σ)$ consisting of locally constant compactly supported $R$-valued functions on $G$, with convolution and involution "twisted" by $σ$. We also introduce a "discretised" analogue of a twist $Σ$ over a Hausdorff étale groupoid $G$, and we show…
We study when the twisted groupoid Banach $*$-algebra $L^1(\mathcal{G},σ)$ is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for $L^1(\mathcal{G},σ)$ to be Hermitian it is sufficient that $L^1 (\mathcal{G}_σ)$ is Hermitian. Moreover, if $\mathcal{G}$ is ample, we find necessary conditions for $L^1(\mathcal{G},σ)$ to be Hermitian in terms of the fibers $\mathcal{G}^x_x$.
We study the groupoid C*-algebras associated to the equivalence relation induced by a quotient map on a locally compact Hausdorff space. This C*-algebra is always a Fell algebra, and if the quotient space is Hausdorff, it is a continuous-trace algebra. We show that the C*-algebra of a locally compact, Hausdorff and principal groupoid is a Fell algebra if and only if the groupoid is one of these relations, extending a theorem of Archbold and Somerset about etale groupoids. The C*-algebras of these relations are, up to Morita equivalence, precisely the Fell algebras with trivial Dixmier-Douady i…
