[Paper Review] Generalized gauge actions on $k$-graph $C^*$-algebras: KMS states and Hausdorff structure

Carla Farsi, Elizabeth Gillaspy+2 more|arXiv (Cornell University)|Jul 23, 2018

Advanced Operator Algebra Research6 references4 citations

TL;DR

This paper establishes a deep connection between KMS states on $k$-graph $C^*$-algebras under generalized gauge actions and the Hausdorff measure and dimension of associated ultrametric Cantor sets. Using a functor $y: \Lambda \to \mathbb{R}_+$, the authors show that KMS states are parametrized by the periodicity group and Borel probability measures arising as Hausdorff measures, with the inverse temperature $\beta$ linked to the Hausdorff dimension $\theta$ via $\beta = \theta$. The key result is that the Hausdorff measure $H^\theta$ coincides with the KMS state measure $\mu_{y,\theta}$, and $\theta$ is the Hausdorff dimension of $\Lambda^\infty$ equipped with a metric derived from $y$.

ABSTRACT

For a finite, strongly connected $k$-graph $Λ$, an Huef, Laca, Raeburn and Sims studied the KMS states associated to the preferred dynamics of the $k$-graph $C^*$-algebra $C^*(Λ)$. They found that these KMS states are determined by the periodicity of $Λ$ and a certain Borel probability measure $M$ on the infinite path space $Λ^\infty$ of $Λ$. Here we consider different dynamics on $C^*(Λ)$, which arise from a functor $y: Λ o \mathbb{R}_+$ 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 $Λ$ and a family of Borel probability measures on the infinite path space. Indeed, these measures also arise as Hausdorff measures on $Λ^\infty$, 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 $y: Λ o \mathbb{R}_+$; the stationary $k$-Bratteli diagram associated to $Λ$; and the concept of exponentially self-similar weights on Bratteli diagrams.

Motivation & Objective

• Understand the structure of KMS states for generalized gauge actions on $k$-graph $C^*$-algebras, extending prior work on the standard gauge action.
• Characterize the inverse temperatures $\beta$ for which KMS states exist under McNamara's generalized gauge dynamics.
• Establish a precise link between the existence of KMS states and the Hausdorff dimension of the infinite path space $\Lambda^\infty$.
• Show that the KMS state measures arise as Hausdorff measures associated with an ultrametric on $\Lambda^\infty$ induced by the functor $y: \Lambda \to \mathbb{R}_+$.
• Prove that the Hausdorff dimension $\theta$ of the ultrametric Cantor set $\Lambda^\infty$ equals the inverse temperature $\beta$ at which KMS states exist.

Proposed method

• The paper constructs an ultrametric $d_{y,\theta}$ on the infinite path space $\Lambda^\infty$ using a functor $y: \Lambda \to \mathbb{R}_+$ and a self-similar weight $w_{y,\theta}$ on the stationary $k$-Bratteli diagram associated to $\Lambda$.
• Using the weight $w_{y,\theta}(\lambda) = e^{-\theta y(\lambda)} \rho(B(y,\theta))^{-d(\lambda)} \xi^{y,\theta}_{s(\lambda)}$, the authors define a metric that induces a self-similar structure on $\Lambda^\infty$.
• By applying the theory of Hausdorff measures on ultrametric Cantor sets, the authors compute the Hausdorff measure $H^\theta$ on cylinder sets $Z(\lambda)$, showing $H^\theta(Z(\lambda)) = \epsilon(\lambda)^\theta$.
• The construction relies on the exponential self-similarity of the weights $w_{y,\theta}$, which ensures the existence of a well-defined Hausdorff dimension $\theta$.
• Using the groupoid model of $C^*$-algebras and quasi-invariant measures, the authors identify the KMS states as being parametrized by Borel probability measures on $\Lambda^\infty$ that are invariant under the dynamics.
• Finally, the paper proves that the KMS state measure $\mu_{y,\theta}$ coincides with the Hausdorff measure $H^\theta$, establishing a direct measure-theoretic correspondence.

Experimental results

Research questions

• RQ1 What are the inverse temperatures $\beta$ for which KMS states exist under generalized gauge actions on $k$-graph $C^*$-algebras?
• RQ2 How are the KMS states parametrized in terms of measures on the infinite path space $\Lambda^\infty$?
• RQ3 What is the geometric and measure-theoretic structure of the infinite path space $\Lambda^\infty$ under the generalized gauge dynamics?
• RQ4 How is the Hausdorff dimension of $\Lambda^\infty$ related to the inverse temperature $\beta$ of the KMS states?
• RQ5 Can the KMS state measures be identified with a canonical Hausdorff measure on $\Lambda^\infty$ equipped with a metric derived from the functor $y: \Lambda \to \mathbb{R}_+$?
• RQ6 Is the Hausdorff dimension $\theta$ of the ultrametric Cantor set $\Lambda^\infty$ equal to the inverse temperature $\beta$ at which KMS states exist?

Key findings

• The inverse temperatures $\beta$ for which KMS states exist under the generalized gauge action are precisely those for which the Hausdorff dimension $\theta$ of $\Lambda^\infty$ equals $\beta$, under the metric $d_{y,\theta}$.
• The KMS states are parametrized by the periodicity group of $\Lambda$ and a family of Borel probability measures on $\Lambda^\infty$, which are shown to be exactly the Hausdorff measures $H^\theta$.
• The Hausdorff dimension $\theta$ of the ultrametric Cantor set $(\Lambda^\infty, d_{y,\theta})$ is equal to the exponent of the self-similar weight $w_{y,\theta}$, and this $\theta$ is the critical inverse temperature for the existence of KMS states.
• The Hausdorff measure $H^\theta$ on $\Lambda^\infty$ agrees exactly with the KMS state measure $\mu_{y,\theta}$, so $H^\theta(Z(\lambda)) = \mu_{y,\theta}(Z(\lambda)) = \epsilon(\lambda)^\theta$ for all cylinder sets $Z(\lambda)$.
• Under the conditions (w-I) or (w-II), the Hausdorff dimension of $\Lambda^\infty$ with respect to the metric $d_{y,\theta}$ is $\theta$, and this dimension governs the existence of KMS states.
• The construction of the metric $d_{y,\theta}$ via the functor $y: \Lambda \to \mathbb{R}_+$ and the associated self-similar weights ensures that the resulting space is an ultrametric Cantor set with well-defined Hausdorff structure.