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.
DOI: http://dx.doi.org/10.7900/jot.2020aug04.2328
Keywords: $C^*$-algebras, Rokhlin property, Rokhlin dimension, Jiang--Su algebra
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