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Symmetries of Kirchberg Algebras


Published:20031201
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]
