The automorphism group of a simple 𝒵-stable 𝒞*-algebra

Ng, Ping; Ruiz, Efren

doi:10.1090/S0002-9947-2013-05728-5

The automorphism group of a simple $\mathcal {Z}$-stable $C^{*}$-algebra
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by Ping Wong Ng and Efren Ruiz PDF

Trans. Amer. Math. Soc. 365 (2013), 4081-4120 Request permission

Abstract:

We study the automorphism group of a simple, unital, $\mathcal {Z}$-stable $C^{*}$-algebra. We show that $\overline {\mathrm {Inn}}_{0} ( \mathfrak {A} )$ is a simple topological group and $\frac { \overline { \mathrm {Inn} } ( \mathfrak {A} ) } {\overline {\mathrm {Inn}}_{0} ( \mathfrak {A} ) }$ is isomorphic (as topological groups) to the inverse limit of quotient groups of $K_{1} (\mathfrak {A} )$, where $\mathfrak {A}$ is a $\mathcal {Z}$-stable $C^{*}$-algebra satisfying the following property: for every UHF algebra $\mathfrak {B}$, $\mathfrak {A} \otimes \mathfrak {B}$ is a nuclear, separable, simple, tracially AI algebra satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet. By the recent results of Lin and Winter, ordered $K$-theory, traces, and the class of the unit is a complete isomorphism invariant for this class of $C^{ *}$-algebra.

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