Equivariant higher Dixmier-Douady theory for circle actions on UHF-algebras

Abstract

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of C⁎-algebras with fibre D⊗K equipped with a fibrewise T-action, where T denotes the circle group and D=End(V)⊗∞ for a T-representation V. In particular, we show that the group of T-equivariant ⁎-automorphisms AutT(D⊗K) is an infinite loop space giving rise to a cohomology theory ED,T⁎(X). Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to ED,T1(X)≅[X,BAutT(D⊗K)]. We compute this group for tori and compare the case D=C to the equivariant Brauer group for trivial actions on the base space.