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Paper IPM / M / 17267

School of Mathematics
Title:	Finite group and integer actions on simple tracially Z-absorbing $C^{*}$ -algebras
Author(s):	Nasser Golestani (Joint with M. Amini, S. Jamali, and N. Christopher Phillips)
Status:	To Appear
Journal:	J. Operator Theory
Supported by:	IPM

Abstract:

We show that if

$A$ is a simple (not necessarily unital) tracially

$\mathcal{Z}$ -absorbing C*-algebra and

$\alpha \colon G \to \mathrm{Aut} (A)$ is an action of a finite group

$G$ on

$A$ with the weak tracial Rokhlin property, then the crossed product

$C^*(G, A,\alpha)$ and the fixed point algebra

$A^\alpha$ are simple and tracially

$\mathcal{Z}$ -absorbing, and they are

$\mathcal{Z}$ -stable if, in addition,

$A$ is separable and nuclear. The same conclusion holds for all intermediate C*-algebras of the inclusions

$A^\alpha \subseteq A$ and

$A \subseteq C^*(G, A,\alpha)$ . We prove that if

$A$ is a simple tracially

$\mathcal{Z}$ -absorbing C*-algebra, then, under a finiteness condition, the permutation action of the symmetric group

$S_m$ on the minimal

$m$ -fold tensor product of

$A$ has the weak tracial Rokhlin property. We define the weak tracial Rokhlin property for automorphisms of simple C*-algebras and we show that---under a mild assumption---(tracial)

$\mathcal{Z}$ -absorption is preserved under crossed products by such automorphisms.

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