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 $D \otimes \mathbb{K}$ equipped with a fibrewise $\mathbb{T}$-action, where $\mathbb{T}$ denotes the circle group and $D = \operatorname{End}\left(V\right)^{\otimes \infty}$ for a $\mathbb{T}$-representation $V$. In particular, we show that the group of $\mathbb{T}$-equivariant $*$-automorphisms $\operatorname{Aut}_{\mathbb{T}}(D \otimes \mathbb{K})$ is an infinite loop space giving rise to a cohomology theory $E^*_{D,\mathbb{T}}(X)$. Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to $E^1_{D,\mathbb{T}}(X) \cong [X, B\operatorname{Aut}_{\mathbb{T}}(D \otimes \mathbb{K})]$. We compute this group for tori and compare the case $D = \mathbb{C}$ to the equivariant Brauer group for trivial actions on the base space.