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Register a new userLimits of groupoid C*-algebras arising from open covers
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I. Raeburn and J. Taylor have constructed continuous-trace C*-algebras with a prescribed Dixmier-Douady class, which also depend on the choice of an open cover of the spectrum. We study the asymptotic behavior of these algebras with respect to certain refinements of the cover and appropriate extension of cocycles. This leads to the analysis of a limit groupoid G and a cocycle sigma, and the algebra C*(G, sigma) may be regarded as a generalized direct limit of the Raeburn-Taylor algebras. As a special case, all UHF C*-algebras arise from this limit construction.
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From a suitable groupoid G, we show how to construct an amenable principal groupoid whose C*-algebra is a Kirchberg algebra which is KK-equivalent to C*(G). Using this construction, we show by example that many UCT Kirchberg algebras can be realised as the C*-algebras of amenable principal groupoids.
In this paper we study the C*-envelope of the (non-self-adjoint) tensor algebra associated via subproduct systems to a finite irreducible stochastic matrix $P$. Firstly, we identify the boundary representations of the tensor algebra inside the Toeplitz algebra, also known as its non-commutative Choquet boundary. As an application, we provide examples of C*-envelopes that are not *-isomorphic to either the Toeplitz algebra or the Cuntz-Pimsner algebra. This characterization required a new proof for the fact that the Cuntz-Pimsner algebra associated to $P$ is isomorphic to $C(mathbb{T}, M_d(mathbb{C}))$, filling a gap in a previous paper. We then proceed to classify the C*-envelopes of tensor algebras of stochastic matrices up to *-isomorphism and stable isomorphism, in terms of the underlying matrices. This is accomplished by determining the K-theory of these C*-algebras and by combining this information with results due to Paschke and Salinas in extension theory. This classification is applied to provide a clearer picture of the various C*-envelopes that can land between the Toeplitz and the Cuntz-Pimsner algebras.
In this paper we show that for an almost finite minimal ample groupoid $G$, its reduced $mathrm{C}^*$-algebra $C_r^*(G)$ has real rank zero and strict comparison even though $C_r^*(G)$ may not be nuclear in general. Moreover, if we further assume $G$ being also second countable and non-elementary, then its Cuntz semigroup ${rm Cu}(C_r^*(G))$ is almost divisible and ${rm Cu}(C_r^*(G))$ and ${rm Cu}(C_r^*(G)otimes mathcal{Z})$ are canonically order-isomorphic, where $mathcal{Z}$ denotes the Jiang-Su algebra.
Renault proved in 2008 that if $G$ is a topologically principal groupoid, then $C_0(G^{(0)})$ is a Cartan subalgebra in $C^*_r(G, Sigma)$ for any twist $Sigma$ 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 associated to these Cartan pairs, under mild additional hypotheses.
We characterise, in several complementary ways, etale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property, assuming that the groupoid is either Hausdorff or $sigma$-compact. This leads directly to a characterisation of the simplicity of this C*-algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C*-algebra. Specifically, we prove that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that is equivalent to the uniqueness of equivariant pseudo-expectations and in the minimal case to an appropriate generalisation of Powers averaging property. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras that includes the essential groupoid C*-algebra itself. By considering a relative version of Powers averaging property, we obtain new examples of C*-irreducible inclusions in the sense of R{o}rdam. These arise from the inclusion of the C*-algebra generated by a suitable group representation into a simple groupoid C*-algebra. This is illustrated by the example of the C*-algebra generated by the quasi-regular representation of Thompsons group T with respect to Thompsons group F, which is contained C*-irreducibly in the Cuntz algebra $mathcal{O}_2$.
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