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Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras

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Abstract

Let (X, Γ) be a free minimal dynamical system, where X is a compact separable Hausdorff space and Γ is a discrete amenable group. It is shown that, if (X, Γ) has a version of Rokhlin property (uniform Rokhlin property) and if C(X) ⋊ Γ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra C(X) ⋊ Γ is at most half of the mean topological dimension of (X, Γ).

These two conditions are shown to be satisfied if Γ = ℤ or if (X, Γ) is an extension of a free Cantor system and Γ 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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Authors and Affiliations

  1. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Zhuang Niu

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  1. Zhuang Niu
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Correspondence to Zhuang Niu.

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The research is partially supported by a Simons Collaboration Grant (Grant #317222) and partially supported by an NSF grant (DMS-1800882).

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Niu, Z. Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras. JAMA 146, 595–672 (2022). https://doi.org/10.1007/s11854-022-0205-8

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  • DOI: https://doi.org/10.1007/s11854-022-0205-8

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