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Journal of Operator Theory

Volume 73, Issue 2, Spring 2015  pp. 465-490.

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

Authors:  Elizabeth Gillaspy
Author institution:Department of Mathematics, University of Colorado - Boulder, Boulder, CO 80309-0395, U.S.A.

Summary: 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 $C^*$-algebras: \[K_*(C^*_\mathrm r(\mathcal{G}, \omega_0)) \cong K_*(C^*_\mathrm r(\mathcal{G}, \omega_1)).\] Generalizing work of S. Echterhoff, W. L\"uck, N.C. Phillips, S. Walters, \textit{J. Reine Angew. Math.} \textbf{639}(2010), 173--221, we show that if $\G = G \ltimes X$ is a transformation group such that $G$ satisfies the Baum--Connes conjecture with coefficients, a homotopy $\omega = \{\omega_t\}_{t \in [0,1]}$ of 2-cocycles on $G \ltimes X$ gives rise to an isomorphism \[K_*(C^*_\mathrm r(G \ltimes X, \omega_0)) \cong K_*(C^*_\mathrm r(G \ltimes X, \omega_1)).\]

Keywords: transformation group, twisted groupoid $C^*$-algebra, $K$-theory, groupoid, $2$-cocycle

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