$K$-theory and homotopies of 2-cocycles on higher-rank graphs

Elizabeth Gillaspy

Vol. 278 (2015), No. 2, 407–426

DOI: 10.2140/pjm.2015.278.407

Abstract

This paper continues our investigation into the question of when a homotopy of 2-cocycles on a locally compact Hausdorff groupoid gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^{\ast }$-algebras. Our main result, which builds on work by Kumjian, Pask, and Sims, shows that a homotopy of 2-cocycles on a row-finite higher-rank graph $\Lambda$ gives rise to twisted groupoid $C^{\ast }$-algebras with isomorphic $K$-theory groups. (The groupoid in question is the path groupoid of $\Lambda$.) We also establish a technical result: any homotopy of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an upper semicontinuous bundle of $C^{\ast }$-algebras.

Keywords

higher-rank graph, twisted groupoid $C^*$-algebra, $K$-theory, twisted $k$-graph $C^*$-algebra, upper semicontinuous $C^*$-bundle, $C_0(X)$-algebra, groupoid, 2-cocycle

Mathematical Subject Classification 2010

Primary: 46L05, 46L80

Milestones

Received: 2 April 2014

Revised: 25 March 2015

Accepted: 18 May 2015

Published: 6 October 2015