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

Abstract

Let Λ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of C⁎(Λ) on certain separable Hilbert spaces of the form L2(X,μ), by introducing the notion of a Λ-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, if Λ is aperiodic, we obtain a faithful representation of C⁎(Λ) on L2(Λ∞,M), where M is the Perron–Frobenius probability measure on the infinite path space Λ∞ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a Λ-semibranching function system gives rise to KMS states for C⁎(Λ). For the higher-rank graphs of Robertson and Steger, we also obtain a representation of C⁎(Λ) on L2(X,μ), where X is a fractal subspace of [0,1] by embedding Λ∞ into [0,1] as a fractal subset X of [0,1]. Moreover, when the Radon–Nikodym derivatives of a Λ-semibranching function system are constant, we show that we can associate to it a KMS state for C⁎(Λ). Finally, we construct a wavelet system for L2(Λ∞,M) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.