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

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.