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

Abstract

This paper continues our investigation into the question of when a homotopy $\omega =\{{\omega }_t{\}}_{t\in [0,1]}$ of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^{\ast }$-algebras: $K_{\ast }(C^{\ast }(\mathcal{G},{\omega }_0))\cong K_{\ast }(C^{\ast }(\mathcal{G},{\omega }_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G}={\mathcal{G}}_{\Lambda }$ is the infinite path groupoid associated to a row-finite higher-rank graph $\Lambda$ with no sources, and $\{c_t{\}}_{t\in [0,1]}$ is a homotopy of 2-cocycles on $\Lambda$, then $K_{\ast }(C^{\ast }({\mathcal{G}}_{\Lambda },{\sigma }_{c_0}))\cong K_{\ast }(C^{\ast }({\mathcal{G}}_{\Lambda },{\sigma }_{c_1})),$ where ${\sigma }_{c_t}$ denotes the 2-cocycle on ${\mathcal{G}}_{\Lambda }$ associated to the 2-cocycle $c_t$ on $\Lambda$. We also prove a technical result (Theorem 3.3), namely that a homotopy of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an upper semi-continuous $C^{\ast }$-bundle.

v2: Exposition condensed. This version to appear in the Pacific Journal of Mathematics

References in corpus (2)

• A Dixmier-Douady classification for Fell algebras
• Groupoid Crossed Products

Cited by in corpus (4)

• Twisted Steinberg algebras
• $\mathrm{K}$-theory and homotopies of twists on ample groupoids
• Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras
• Self-similar groupoid actions on k-graphs, and invariance of K-theory for cocycle homotopies