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Symmetries of Kirchberg Algebras

Published online by Cambridge University Press:  20 November 2018

David J. Benson ,
Alex Kumjian  and
N. Christopher Phillips
David J. Benson
Affiliation:
Department of Mathematics University of Georgia Athens, GA 30602-7403 USA, email: djb@byrd.math.uga.edu
Alex Kumjian
Affiliation:
Department of Mathematics University of Nevada Reno, NV 89557-0045 USA, email: alex@unr.edu
N. Christopher Phillips
Affiliation:
Department of Mathematics University of Oregon Eugene, OR 97403-1222 USA, email: ncp@darkwing.uoregon.edu
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Abstract

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Let ${{G}_{0}}$ and ${{G}_{1}}$ be countable abelian groups. Let ${{\gamma }_{i}}$ be an automorphism of ${{G}_{i}}$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[{{1}_{A}}]\,=\,0$ in ${{K}_{0}}(A)$, and an automorphism $\alpha \,\in \,\text{Aut}(A)$ of order two, such that ${{K}_{0}}(A)\,\cong \,{{G}_{0}}$, such that ${{K}_{1}}(A)\,\cong \,{{G}_{1}}$, and such that ${{\alpha }_{*}}\,:\,{{K}_{i}}(A)\,\to \,{{K}_{i}}(A)$ is ${{\gamma }_{i}}$. As a consequence, we prove that every ${{\mathbb{Z}}_{2}}$-graded countable module over the representation ring $R({{\mathbb{Z}}_{2}})$ of ${{\mathbb{Z}}_{2}}$ is isomorphic to the equivariant $K$-theory ${{K}^{{{\mathbb{Z}}_{2}}}}(A)$ for some action of ${{\mathbb{Z}}_{2}}$ on a unital Kirchberg algebra $A$.

Along the way, we prove that every not necessarily finitely generated $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$ itself and $\mathbb{Z}$ on which the nontrivial element of ${{\mathbb{Z}}_{2}}$ acts either trivially or by multiplication by −1.

Keywords

20C1046L5519K9919L4746L4046L80
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 4 , 01 December 2003 , pp. 509 - 528
Copyright
Copyright © Canadian Mathematical Society 2003

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