The automorphism group of a simple $\mathcal {Z}$-stable $C^{*}$-algebra
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- by Ping Wong Ng and Efren Ruiz PDF
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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.References
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Additional Information
- Ping Wong Ng
- Affiliation: Department of Mathematics, University of Louisiana at Lafayette, 217 Maxim D. Doucet Hall, P.O. Box 41010, Lafayette, Louisiana 70504-1010
- MR Author ID: 699995
- Email: png@louisiana.edu
- Efren Ruiz
- Affiliation: Department of Mathematics, University of Hawaii at Hilo, 200 W. Kawili Street, Hilo, Hawaii 96766
- MR Author ID: 817213
- Email: ruize@hawaii.edu
- Received by editor(s): October 26, 2010
- Received by editor(s) in revised form: September 7, 2011
- Published electronically: March 11, 2013
- Additional Notes: The authors are grateful to the referee for a careful reading of the paper and useful suggestions.
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 4081-4120
- MSC (2010): Primary 46L35
- DOI: https://doi.org/10.1090/S0002-9947-2013-05728-5
- MathSciNet review: 3055690