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.

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In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph Λ, via the infinite path space Λ ∞ of Λ. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of Λ ∞ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary -Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph Λ. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are -regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure , and show that is a rescaled version of the measure on Λ ∞ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of 2 (Λ ∞ , ) which was constructed by Farsi et al. 2.5 below. The space of infinite paths  of a stationary -Bratteli diagram  is often a Cantor set, enabling us to study its associated Pearson-Bellissard spectral triple. Indeed, if the matrices 1 , … , are the adjacency matrices for a -graph Λ, then the space of infinite paths in Λ is homeomorphic to the Cantor set  (also called ). In other words, the Pearson-Bellissard spectral triples for stationary -Bratteli diagrams can also be viewed as spectral triples for higher-rank graphs.We then proceed to study, in Section 3, the geometrical information encoded by these spectral triples. Theorem 3.14 establishes that the Pearson-Bellissard spectral triple associated to (  Λ , ) is finitely summable, with dimension ∈ (0, 1). Section 3.3 focuses on the Dixmier traces of the spectral triples, and establishes both an integral formula for the Dixmier trace (Theorems 3.23 and 3.28) and a concrete expression for the measure induced by the Dixmier trace (Theorem 3.26). These computations also reveal that the ultrametric Cantor sets (  Λ , ) are -regular in the sense of [59, Definition 11]. Other settings in the literature in which spectral triples on Cantor sets admit an integral formula for the Dixmier trace include [13,47,17,14].In full generality, Dixmier traces are defined on the Dixmier-Macaev (also called Lorentz) ideal  1,∞ ⊆ () inside the compact operators and are computed using a generalized limit (roughly speaking, a linear functional that lies between lim sup and lim inf). Although the theory of Dixmier traces can be quite intricate, many of the computations simplify substantially in our setting, and so our treatment of the general theory will be brief; we refer the interested reader to the extensive literature on Dixmier traces and other singular traces (cf. [19, 55, 54, 1...

We describe the primitive ideal space of the C * -algebra of a row-finite k-graph with no sources when every ideal is gauge invariant. We characterize which spectral spaces can occur, and compute the primitive ideal space of two examples. In order to do this we prove some new results on aperiodicity. Our computations indicate that when every ideal is gauge invariant, the primitive ideal space only depends on the 1-skeleton of the k-graph in question.

Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

In this article we provide an identification between the wavelet decompositions of certain fractal representations of C * algebras of directed graphs of M. Marcolli and A. Paolucci [19], and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of Cuntz C * -algebras O D . In particular, in this setting we use results of J. Pearson and J. Bellissard [20], and A. Julien and J. Savinien [15], to construct first the spectral triple and then the Laplace-Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace-Beltrami operators for the equal weight case for representations of O D , and in the uneven weight case for certain representations of O 2 , and show how the eigenspaces and wavelet subspaces at different levels first constructed in [8] are related.2010 Mathematics Subject Classification: 46L05.

In this paper we define the notion of monic representation for the C * -algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative C * -algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the Λ-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

We study the equilibrium or KMS states of the Toeplitz C * -algebra of a finite higher-rank graph which is reducible. The Toeplitz algebra carries a gauge action of a higher-dimensional torus, and a dynamics arises by choosing an embedding of the real numbers in the torus. Here we use an embedding which leads to a dynamics which has previously been identified as "preferred", and we scale the dynamics so that 1 is a critical inverse temperature. As with 1-graphs, we study the strongly connected components of the vertices of the graph. The behaviour of the KMS states depends on both the graphical relationships between the components and the relative size of the spectral radii of the vertex matrices of the components.We test our theorems on graphs with two connected components. We find that our techniques give a complete analysis of the KMS states with inverse temperatures down to a second critical temperature β c < 1.