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Register a new userStrict comparison for $C^*$-algebras arising from Almost finite groupoids
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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.
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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.
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.
Let $G$ be a Hausdorff, etale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, ample groupoid, then we show that $C^*_r(G)$ is purely infinite simple if and only if every nonzero projection in $C_0(G^0)$ is infinite in $C^*_r(G)$. We then show how this result applies to $k$-graph $C^*$-algebras. Finally, we investigate strongly purely infinite groupoid $C^*$-algebras.
Given a normal subgroup bundle $mathcal A$ of the isotropy bundle of a groupoid $Sigma$, we obtain a twisted action of the quotient groupoid $Sigma/mathcal A$ on the bundle of group $C^*$-algebras determined by $mathcal A$ whose twisted crossed product recovers the groupoid $C^*$-algebra $C^*(Sigma)$. Restricting to the case where $mathcal A$ is abelian, we describe $C^*(Sigma)$ as the $C^*$-algebra associated to a $mathbf T$-groupoid over the tranformation groupoid obtained from the canonical action of $Sigma/mathcal A$ on the Pontryagin dual space of $mathcal A$. We give some illustrative examples of this result.
In recent work, Cuntz, Deninger and Laca have studied the Toeplitz type C*-algebra associated to the affine monoid of algebraic integers in a number field, under a time evolution determined by the absolute norm. The KMS equilibrium states of their system are parametrized by traces on the C*-algebras of the semidirect products $J rtimes O^*$ resulting from the multiplicative action of the units $O^*$ on integral ideals $J$ representing each ideal class. At each fixed inverse temperature $beta > 2$, the extremal equilibrium states correspond to extremal traces of $C^*(Jrtimes O^*)$. Here we undertake the study of these traces using the transposed action of $O^*$ on the duals $hat J$ of the ideals and the recent characterization of traces on transformation group C*-algebras due to Neshveyev. We show that the extremal traces of $C^*(Jrtimes O^*)$ are parametrized by pairs consisting of an ergodic invariant measure for the action of $O^*$ on $hat{J}$ together with a character of the isotropy subgroup associated to the support of this measure. For every ideal the dual group $hat {J}$ is a d-torus on which $O^*$ acts by linear toral automorphisms. Hence, the problem of classifying all extremal traces is a generalized version of Furstenbergs celebrated $times 2$ $times 3$ conjecture. We classify the results for various number fields in terms of ideal class group, degree, and unit rank, and we point along the way the trivial, the intractable, and the conjecturally classifiable cases. At the topological level, it is possible to characterize the number fields for which infinite $O^*$-invariant sets are dense in $hat{J} $, thanks to a theorem of Berend; as an application we give a description of the primitive ideal space of $C^*(Jrtimes O^*)$ for those number fields.
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