The Automorphism group of a simple $\mathcal{Z}$-stable $C^{*}$-algebra

TL;DR

This paper investigates the automorphism group of a certain class of simple, $ ext{Z}$-stable $C^{*}$-algebras, extending previous results and leveraging classification theorems based on $K$-theory and traces.

Contribution

It generalizes earlier work to include a broader class of $ ext{Z}$-stable $C^{*}$-algebras that satisfy specific classification conditions.

Findings

Automorphism groups characterized for the class of algebras considered.

Extension of previous automorphism results to new algebra classes.

Utilization of classification theorems to analyze automorphisms.

Abstract

We study the automorphism group of a unital, simple, $\mathcal{Z}$-stable $C^{*}$-algebra. In this paper, we generalize the results by the authors in \cite{pr_auto} to $\mathcal{Z}$-stable $C^{*}$-algebras $\mathfrak{A}$ such that $\mathfrak{A} \otimes \mathfrak{B}$ is a separable, nuclear, simple, tracially AI algebras satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet \cite{uct}. By the results of Lin in \cite{hl_asyunit} and Winter in \cite{ww_localelliott}, $C^{\ast}$-algebras that satisfies the above condition are classified via $K$-theory and traces.