Crossed products of C⁎C⁎-algebras for singular actions

We consider group actions α:G→Aut(A)α:G→Aut(A) of topological groups G   on C⁎C⁎-algebras AA of the type which occurs in many physics models. These are singular actions in the sense that they need not be strongly continuous, or the group need not be locally compact. We develop a “crossed product host” in analogy to the usual crossed product for strongly continuous actions of locally compact groups, in the sense that its representation theory is in a natural bijection with the covariant representation theory of the action α:G→Aut(A)α:G→Aut(A). We prove a uniqueness theorem for crossed product hosts, and analyze existence conditions. We also present a number of examples where a crossed product host exists, but the usual crossed product does not. For actions where a crossed product host does not exist, we obtain a “maximal” invariant subalgebra for which a crossed product host exists. We further study the case of a discontinuous action α:G→Aut(A)α:G→Aut(A) of a locally compact group in detail.