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Symmetries of Kirchberg Algebras

  Published:2003-12-01
 Printed: Dec 2003
  • David J. Benson
  • Alex Kumjian
  • N. Christopher Phillips
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Abstract

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 $\alpha \in \Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1 (A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ 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 $\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and $\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts either trivially or by multiplication by $-1$.
MSC Classifications: 20C10, 46L55, 19K99, 19L47, 46L40, 46L80 show english descriptions Integral representations of finite groups
Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
None of the above, but in this section
Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]
Automorphisms
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
20C10 - Integral representations of finite groups
46L55 - Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
19K99 - None of the above, but in this section
19L47 - Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91]
46L40 - Automorphisms
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
 

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