Equivariant higher DixmierDouady Theory for circle actions on UHFalgebras
Abstract
We develop an equivariant DixmierDouady 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.13364
 arXiv:
 arXiv:2201.13364
 Bibcode:
 2022arXiv220113364E
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Algebraic Topology;
 46L35;
 46M20;
 55N20
 EPrint:
 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)