$K$-theory and homotopies of 2-cocycles on transformation groups

Abstract

This paper constitutes a first step in the author's program to investigate the question of when a homotopy of 2-cocycles $\omega =\{{\omega }_t{\}}_{t\in [0,1]}$ on a locally compact Hausdorff groupoid $\mathcal{G}$ induces an isomorphism of the $K$-theory groups of the reduced twisted groupoid $C^{\ast }$-algebras: $K_{\ast }(C_r^{\ast }(\mathcal{G},{\omega }_0))\cong K_{\ast }(C_r^{\ast }(\mathcal{G},{\omega }_1)).$ Generalizing work of Echterhoff, Lück, Phillips, and Walters from 2010, we show that if $\mathcal{G}=G⋉X$ is a second countable locally compact transformation group, then whenever $G$ satisfies the Baum-Connes conjecture with coefficients, a homotopy $\omega =\{{\omega }_t{\}}_{t\in [0,1]}$ of 2-cocycles on $G⋉X$ gives rise to an isomorphism $K_{\ast }(C_r^{\ast }(G⋉X,{\omega }_0))\cong K_{\ast }(C_r^{\ast }(G⋉X,{\omega }_1)).$

arXiv commentSome improvements to the exposition; also, the hypotheses on Theorem 5.1 have been relaxed so that X is no longer required to be compact. This version (v3) fixes the erroneous argument in v2 for this strengthening of Theorem 5.1. This is the version that will appear in the Journal of Operator Theory