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

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 * -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 Λ gives rise to twisted groupoid C * -algebras with isomorphic Ktheory groups. (Here, the groupoid in question is the path groupoid of Λ.) We also establish a technical result: any homotopy of 2-cocycles on a locally compact Hausdorff groupoid G gives rise to an upper semicontinuous bundle of C * -algebras.

$K$-theory and homotopies of 2-cocycles on group bundles

“…Proposition 2.9 below describes the C 0 (M )-algebra structure carried by the twisted C * -algebra of a group bundle π : G → M . A similar result is obtained by Goehle for crossed products by a group bundle in Proposition 1.2 and Lemma 1.4 of [6], and the proof of Proposition 2.9 below proceeds along similar lines to Goehle's proof, and also to the proof of Theorem 3.5 from [4].…”

“…Group bundles are examples of groupoids; the results of this article thus continue the author's research program, begun in [5,4], to investigate the question of when a homotopy {ω t } t∈[0,1] of 2-cocycles on a groupoid G induces an isomorphism K * (C * (G, ω 0 )) ∼ = K * (C * (G, ω 1 )) (1) of the K-theory groups of the twisted groupoid C * -algebras. This question was inspired by the realization that Bott periodicity and the noncommutative tori can both be viewed as examples of a K-theoretic isomorphism arising from a homotopy of 2-cocycles.…”

“…In addition to its philosophical links with [5,4], an attentive reader will notice similarities between the proofs presented in this article and several main results from [5,4]. To be precise, we begin this article by following the outline of the proof of Theorem 3.5 from [4] to calculate the C 0 (M )-algebra structure associated to a locally trivial bundle of groups π : G → M . Then we use the results of this calculation, together with Theorem 5.1 from [5] and a Mayer-Vietoris argument, to establish our main result in Theorem 3.3; Corollary 3.4 follows immediately.…”

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

“…In a forthcoming paper [9], we extend the techniques of [19] to show that any homotopy of cocycles on Λ gives rise to K-equivalent twisted higher-rank graph C * -algebras; it has been suggested to us that these techniques might also apply to the case of Deaconu-Renault groupoids.…”

“…). In addition to the transformation-group case G = G ⋉ X considered in the present paper, we will address this question in a forthcoming paper [9] for the case when G = G Λ is the groupoid associated to a higher-rank graph Λ. (A generalization of directed graphs, higher-rank graphs and their associated groupoids and C *algebras were introduced in [18].…”