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

Abstract

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of $C^{\ast }$-algebras with fibre $D\otimes \mathbb{K}$ equipped with a fibrewise $\mathbb{T}$-action, where $\mathbb{T}$ denotes the circle group and $D=\mathrm{End}{\left(V\right)}^{\otimes \infty }$ for a $\mathbb{T}$-representation $V$. In particular, we show that the group of $\mathbb{T}$-equivariant $\ast$-automorphisms ${\mathrm{Aut}}_{\mathbb{T}}(D\otimes \mathbb{K})$ is an infinite loop space giving rise to a cohomology theory $E_{D,\mathbb{T}}^{\ast }(X)$. Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to $E_{D,\mathbb{T}}^1(X)\cong [X,B{\mathrm{Aut}}_{\mathbb{T}}(D\otimes \mathbb{K})]$. 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.

37 pages, published version (except for a typo in the description of the order structure on page 19 and the proof of Lemma 3.5, which was fixed after publication, and did not change the main result)

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