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

Abstract

We study the automorphism group of a unital, simple, $\mathcal{Z}$-stable $C^{\ast }$-algebra. In this paper, we generalize the results by the authors in \cite{pr_auto} to $\mathcal{Z}$-stable $C^{\ast }$-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.

Improved the exposition of Section 3 and Section 4, added other references