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

Abstract

Let $\Lambda$ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of $C^{\ast }(\Lambda )$ 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 $C^{\ast }(\Lambda )$ 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 $C^{\ast }(\Lambda )$. For the higher-rank graphs of Robertson and Steger, we also obtain a representation of $C^{\ast }(\Lambda )$ 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 $C^{\ast }(\Lambda )$ 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.

Modified hypotheses in Theorem 3.5