Separable representations, KMS states, and wavelets for higher-rank graphs

TL;DR

This paper develops new representations and KMS states for higher-rank graph C*-algebras, introduces wavelet systems, and explores fractal embeddings, advancing understanding of their structure and states.

Contribution

It constructs novel separable Hilbert space representations, establishes KMS states linked to fractal dimensions, and generalizes wavelet systems to higher-rank graphs.

Findings

Faithful representation on $L^2( ext{infinite path space})$ for aperiodic graphs

Existence of KMS states with inverse temperature equal to Hausdorff dimension

Wavelet systems constructed for higher-rank graph C*-algebras

Abstract

Let $\Lambda$ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of $C^*(\Lambda)$ on certain separable Hilbert spaces of the form $L^2(X,\mu)$, by introducing the notion of a $\Lambda$-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, when $\Lambda$ is aperiodic, we obtain a faithful representation of $C^*(\Lambda)$ on $L^2(\Lambda^\infty, M)$, where $M$ is the Perron-Frobenius probability measure on the infinite path space $\Lambda^\infty$ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a $\Lambda$-semibranching function system gives rise to KMS states for $C^*(\Lambda)$. For the higher-rank graphs of Robertson and Steger, we also obtain a representation of $C^*(\Lambda)$ on $L^2(X, \mu)$, where $X$ is a fractal subspace of $[0,1]$ by embedding $\Lambda^{\infty}$ into $[0,1]$ as a fractal subset $X$ of $[0,1]$. In this latter case we additionally show that there exists a KMS state for $C^*(\Lambda)$ whose inverse temperature is equal to the Hausdorff dimension of $X$. Finally, we construct a wavelet system for $L^2(\Lambda^\infty, M)$ by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.