Deformation of operator algebras by Borel cocycles

TL;DR

This paper introduces a deformation of C*-algebras using Borel cocycles, showing invariance of K-theory under certain conditions and extending previous results in operator algebra theory.

Contribution

It defines a new deformation A_y Borel cocycles and proves K-theory invariance for groups satisfying the Baum-Connes conjecture.

Findings

A_an be expressed via crossed products involving G.

K-theory of A_ re invariant under homotopies of or certain groups.

Extension of existing results on operator algebra deformations.

Abstract

Assume that we are given a coaction \delta of a locally compact group G on a C*-algebra A and a T-valued Borel 2-cocycle \omega on G. Motivated by the approach of Kasprzak to Rieffel's deformation we define a deformation A_\omega of A. Among other properties of A_\omega we show that A_\omega\otimes K(L^2(G)) is canonically isomorphic to A\rtimes_\delta\hat G\rtimes_{\hat\delta,\omega}G. This, together with a slight extension of a result of Echterhoff et al., implies that for groups satisfying the Baum-Connes conjecture the K-theory of A_\omega remains invariant under homotopies of omega.