Symmetries of Kirchberg Algebras

Abstract

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