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Register a new userThe classification of Rokhlin flows on C*-algebras
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Authors
Gabor Szabo
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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.
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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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