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

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of C-*-algebras with fibre D circle times K equipped with a fibrewise T-action, where T denotes the circle group and D = End (V)(circle times infinity) for a T-representation V. In particular, we show that the group of T-equivariant *- automorphisms AutT (D circle times K) is an infinite loop space giving rise to a cohomology theory E-D,T(*) (X). Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to E-D,T(1) (X) similar or equal to [X, BAut(T) (D circle times 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. (c) 2022 The Authors. Published by Elsevier Inc.