Cocycles on Certain Groupoids Associated to ${\mathbb{N}}^k$-Actions

Abstract

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $C^{\ast }$-algebras associated to these groupoids. We provide a new characterization of $1$-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $k$-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($k$-Ruelle triples and commuting Ruelle operators). Results on KMS states on $C^{\ast }$-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence-uniqueness results for KMS states associated to the graphs.

34 pages, 1 figure. Substantial changes have been made to Version 1 in order to improve and streamline the exposition

References in corpus (2)

• KMS states on the C*-algebras of non-principal groupoids
• Generalized gauge actions on $k$-graph $C^{\ast }$-algebras: KMS states and Hausdorff structure