Symmetries of Kirchberg Algebras

Abstract

Let G0 and G1 be countable abelian groups. Let γi, 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 α ∈ Aut(A) of order two, such that K0(A) ≅ G 0, such that K1(A) ≅ G1, and such that α* : K1(A) → Ki(A) is γi. As a consequence, we prove that every ℤ2 -graded countable module over the representation ring R(ℤ2 ) of ℤ2 is isomorphic to the equivariant K -theory K ℤ2 (A) for some action of ℤ2 on a unital Kirchberg algebra A. Along the way, we prove that every not necessarily finitely generated ℤ[ℤ2] -module which is free as a ℤ-module has a direct sum decomposition with only three kinds of summands, namely ℤ[ℤ2] itself and ℤ on which the nontrivial element of ℤ2 acts either trivially or by multiplication by - 1.