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A Galois Correspondence for Reduced Crossed Products of Simple $\text{C}^{\ast }$-algebras by Discrete Groups

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  07 January 2019

Jan Cameron  and
Roger R. Smith
Jan Cameron
Affiliation:
Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, USA Email: jacameron@vassar.edu
Roger R. Smith
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA Email: rsmith@math.tamu.edu
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Abstract

Let a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

Keywords

C∗ -algebragroupcrossed productbimodulereducedtwisted

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 46L40: Automorphisms
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 1103 - 1125
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Author J. C. was partially supported by Simons Collaboration Grant for Mathematicians #319001. Author R. S. was partially supported by Simons Collaboration Grant for Mathematicians #522375.

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