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Journal of Operator Theory

Volume 87, Issue 1, Winter 2022  pp. 157-186.

Rokhlin-type properties, approximate innerness and $\mathcal{Z}$-stability

Authors:  Ilan Hirshberg
Author institution: Department of Mathematics, Ben Gurion University of the Negev, Be'er Sheva 84105, Israel

Summary: We investigate connections between actions on separable $C^*$-algebras with Rokhlin-type properties and absorption of the Jiang-Su algebra $\mathcal{Z}$. We show that if $A$ admits an approximately inner group action with finite Rokhlin dimension with commuting towers then $A$ is $\mathcal{Z}$-stable. We obtain analogous results for tracial version of the Rokhlin property and approximate innerness. Going beyond approximate innerness, for actions of a single automorphism which have the Rokhlin property and are almost periodic in a suitable sense, the crossed product absorbs $\mathcal{Z}$ even when the original algebra does not.

Keywords: $C^*$-algebras, Rokhlin property, Rokhlin dimension, Jiang--Su algebra

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