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Register a new userComparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras
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Zhuang Niu
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Let $(X, Gamma)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $Gamma$ is a discrete amenable group. It is shown that, if $(X, Gamma)$ has a version of Rokhlin property (uniform Rokhlin property) and if $mathrm{C}(X)rtimesGamma$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $mathrm{C}(X) rtimes Gamma$ is at most half of the mean topological dimension of $(X, Gamma)$. These two conditions are shown to be satisfied if $Gamma = mathbb Z$ or if $(X, Gamma)$ is an extension of a free Cantor system and $Gamma$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.
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Consider a minimal free topological dynamical system $(X, T, mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, mathbb{Z}^d)$. As a consequence, the C*-algebra $mathrm{C}(X) rtimes mathbb{Z}^d$ is classifiable if $(X, T, mathbb{Z}^d)$ has zero mean dimension.
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.
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 $mathcal{O}_2$.
We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{o}rdams simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.
We consider unital simple inductive limits of generalized dimension drop C*-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop $C^*$-algebras. Suppose that $A$ is one of these C*-algebras. We show that $Aotimes Q$ has tracial rank no more than one, where $Q$ is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let $A$ and $B$ be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then $Acong B$ if and only if they have the same Elliott invariant.
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