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 L 2 (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 ofwhere 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, where X is a fractal subspace of [0, 1] by embedding Λ ∞ 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 * (Λ) whose inverse temperature is equal to the Hausdorff dimension of X. Finally, we construct a wavelet system for L 2 (Λ ∞ , M ) by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.2010 Mathematics Subject Classification: 46L05.

Monic representations of finite higher-rank graphs

“…In [27], separable representations of C * (Λ) were constructed by using Λ-semibranching function systems on measure spaces. A Λ-semibranching function system is a generalization of the semibranching function systems studied by Marcolli and Paolucci in [40].…”

“…In addition to connections with wavelets (cf. [17,18,40,27,28,26]), representations of Cuntz-Krieger algebras have been linked to fractals and Cantor sets [48,35,25,26] and to the endomorphism group of a Hilbert space [8,39]. Indeed, the astonishing goal of identifying both discrete and continuous series of representations of Cuntz (and to some extent Cuntz-Krieger) C * -algebras, was accomplished in [19,20,5], building on the pioneering results of [7].…”

“…The monic representations that we focus on in this paper were inspired in part by the wavelet theory for higher-rank graphs which was developed in [27]. These wavelets relied on the concept of Λ-semibranching function systems, introduced in [27] and further studied in [24]. In this paper, we refine the Λ-semibranching function systems into a crucial technical tool for studying monic representations, namely the Λ-projective systems of Definition 3.1.…”

“…Before turning our attention to a theoretic and systematic analysis of the monic representations of finite k-graph C * -algebras, Section 3 develops the technical tools we will need for this analysis. The Λ-semibranching function systems of [27] are refined in Section 3.1 into Λ-projective systems, and Section 3.2 analyzes the projectionvalued measure P = P π on the infinite path space Λ ∞ which arises from a representation π of C * (Λ). Section 3 ends by addressing the question of when representations of k-graph C * -algebras are disjoint or irreducible, see Theorems 3.11 and 3.13.…”

Dirichlet Forms and Ultrametric Cantor Sets Associated to Higher-Rank Graphs

“…Here φ 1 (λ) and φ 2 (λ) are infinite paths in [λ] and the subscripts 1,2 imply that the choice function gives a pair of (distinct) infinite paths in [λ] satisfying (7). The condition in (7) means that φ 1 (λ) and φ 2 (λ) satisfy φ 1 (λ) ∈ [λe] and φ 2 (λ) ∈ [λe ′ ] for two different edges e, e ′ . According to [17], the space of choice functions is the analogue of the sphere bundle of a Riemannian manifold.…”