Symmetries of Kirchberg algebras

David J.\  Benson, Alex Kumjian, and N.\  Christopher Phillips


\begin{abstract}
Let $G_0$ and $G_1$ be countable abelian groups.
Let $\gm_i$ be an automorphism of $G_i$ \oot.
Then there exists a unital Kirchberg algebra $A$ satisfying the \uct\  %
and with $[1_A] = 0$ in $K_0 (A)$,
and an automorphism $\af \in \Aut (A)$ \oot,
such that $K_0 (A) \cong G_0$, such that $K_1 (A) \cong G_1$, and
such that $\af_* \colon K_i (A) \to K_i (A)$ is $\gm_i$.
As a consequence, we prove that every $\Zt$-graded countable module
over the representation ring $R (\Zt)$ of $\Zt$
is isomorphic to the equivariant K-theory $K^{\Zt} (A)$ for some
action of $\Zt$ on a unital Kirchberg algebra $A$.

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