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Register a new user$mathcal Z$-stability of $mathrm{C}(X)rtimesGamma$
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Authors
Zhuang Niu
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Let $(X, Gamma)$ be a free and minimal topological dynamical system, where $X$ is a separable compact Hausdorff space and $Gamma$ is a countable infinite discrete amenable group. It is shown that if $(X, Gamma)$ has the Uniform Rokhlin Property and Cuntz comparison of open sets, then $mathrm{mdim}(X, Gamma)=0$ implies that $(mathrm{C}(X) rtimesGamma)otimesmathcal Z cong mathrm{C}(X) rtimesGamma$, where $mathrm{mdim}$ is the mean dimension and $mathcal Z$ is the Jiang-Su algebra. In particular, in this case, $mathrm{mdim}(X, Gamma)=0$ implies that the C*-algebra $mathrm{C}(X) rtimesGamma$ is classified by the Elliott invariant.
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It is shown that, for an arbitrary free and minimal $mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $mathrm{C}(X)rtimesmathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, Gamma)$, where $Gamma$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $mathrm{C}(X)rtimesGamma$ has stable rank one. Moreover, in this case, the C*-algebra $mathrm{C}(X)rtimesGamma$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.
We investigate the notion of tracial $mathcal Z$-stability beyond unital C*-algebras, and we prove that this notion is equivalent to $mathcal Z$-stability in the class of separable simple nuclear C*-algebras.
We define a notion of tracial $mathcal{Z}$-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially $mathcal{Z}$-absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.
We study flows on C*-algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C*-algebras satisfying certain technical properties, which hold for many C*-algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimotos conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, strongly purely infinite C*-algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable $KK$-contractible C*-algebras: Two Rokhlin flows on such a C*-algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate.
The representations of a $k$-graph $C^*$-algebra $C^*(Lambda)$ which arise from $Lambda$-semibranching function systems are closely linked to the dynamics of the $k$-graph $Lambda$. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible $Lambda$-semibranching representations and purely atomic representations of $C^*(Lambda)$. Throughout the paper, we work in the setting of row-finite source-free $k$-graphs; this paper constitutes the first analysis of $Lambda$-semibranching representations at this level of generality.
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