Spectral triples and wavelets for higher-rank graphs

Abstract

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...

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

“…Definition 3.1. [17,13,9] An odd spectral triple is a triple (A, H, D) 6 consisting of a Hilbert space H, an involutive algebra A of (bounded) operators on H and a densely defined self-adjoint operator D that has compact resolvent such that [D, π(a)] is a bounded operator for all a ∈ A, where π is a faithful bounded * -representation of A on H. An even spectral triple is an odd spectral triple with a grading operator (meaning self adjoint and unitary) Γ on H such that ΓD = −DΓ, and Γπ(a) = π(a)Γ for all a ∈ A.…”

“…(See Theorem 3.9 and Corollary 3.10 of [9] for the details). Now we describe the associated Dirichlet form and Laplace-Beltrami operator as follows.…”

“…(20) 9 Note that the definition works for an arbitrary tree as long as there exists a measure on the associated infinite path space. See also Definition 10.6 of [14].…”

“…The measure M is often called the Perron-Frobenius measure on Λ ∞ . Now we describe Bratteli diagrams and k-Bratteli diagrams introduced in [2,9] as follows. 3 Since y ∈ Λ ∞ , we have y : Ω k → Λ by definition.…”

“…Definition 2.3. [2,9] A Bratteli diagram denoted by B is a directed graph with a vertex set B 0 = n∈N V n , and an edge set B 1 = ∞ n=1 E n , where E n consists of edges whose source vertex lies in V n+1 and whose range vertex lies in V n . A finite path η = η 1 · · · η ℓ is a finite sequence of edges with r(η n ) = s(η n+1 ).…”