Generalized gauge actions on $k$-graph $C^{\ast }$-algebras: KMS states and Hausdorff structure

Abstract

For a finite, strongly connected $k$-graph $\Lambda$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^{\ast }$-algebra $C^{\ast }(\Lambda )$. They found that these KMS states are determined by the periodicity of $\Lambda$ and a certain Borel probability measure $M$ on the infinite path space ${\Lambda }^{\infty }$ of $\Lambda$. Here we consider different dynamics on $C^{\ast }(\Lambda )$, which arise from a functor $y:\Lambda \to {\mathbb{R}}_{+}$ and were first proposed by McNamara in his thesis. We show that the KMS states associated to McNamara's dynamics are again parametrized by the periodicity group of $\Lambda$ and a family of Borel probability measures on the infinite path space. Indeed, these measures also arise as Hausdorff measures on ${\Lambda }^{\infty }$, and the associated Hausdorff dimension is intimately linked to the inverse temperatures at which KMS states exist. Our construction of the metrics underlying the Hausdorff structure uses the functors $y:\Lambda \to {\mathbb{R}}_{+}$; the stationary $k$-Bratteli diagram associated to $\Lambda$; and the concept of exponentially self-similar weights on Bratteli diagrams.