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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Ping Wong Ng and Efren Ruiz
Journal: Trans. Amer. Math. Soc. 365 (2013), 4081-4120
MSC (2010): Primary 46L35
Published electronically: March 11, 2013
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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.


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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
Email: png@louisiana.edu

Efren Ruiz
Affiliation: Department of Mathematics, University of Hawaii at Hilo, 200 W. Kawili Street, Hilo, Hawaii 96766
Email: ruize@hawaii.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05728-5
PII: S 0002-9947(2013)05728-5
Keywords: Automorphism, topological groups
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.
Article copyright: © Copyright 2013 American Mathematical Society