Symmetries of Kirchberg algebras

Let Go and G, 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 {1A} = 0 in K-0(A), and an automorphism alpha is an element of Aut(A) of order two, such that K-0(A) Go, such that K-1(A) congruent to G(1), and such that a(*) : K-i(A) --> K-i(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-Z2 (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.