Symmetries of Kirchberg Algebras

Let G0 and G1 be countable abelian groups. Let (gamma)i be an automorphism of Gi of order two. Then there exists a unital Kirchberg algebra A satisfying the Universal Coefficient Theorem and with [1A] = 0 in K0 (A), and an automorphism (alpha)(is a member of) \Aut(A) of order two, such that K0 (A) =~G0, such that K1 (A) =~G1, and such that (alpha)* : Ki (A) -Ki (A) is (gamma)i. As a consequence, we prove that every {Z}2-graded countable module over the representation ring R ({Z}2) of {Z}2 is isomorphic to the equivariant K-theory K\{Z}2 (A) for some action of {Z}2 on a unital Kirchberg algebra A. Along the way, we prove that every not necessarily finitely generated {Z} [{Z}2]module which is free as a {Z}-module has a direct sum decomposition with only three kinds of summands, namely {Z} [{Z}2] itself and {Z} on which the nontrivial element of {Z}2 acts either trivially or by multiplication by -1.