Spectral triples for higher-rank graph $C^*$-algebras

Abstract

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(\Lambda )$ of a strongly connected finite higher-rank graph Λ. Our spectral triple builds on an approach used by Consani and Marcolli to construct spectral triples for Cuntz-Krieger algebras. We prove that our spectral triples are intimately connected to the wavelet decomposition of the infinite path space of Λ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of our spectral triple. The paper concludes by discussing other properties of the spectral triple, namely, θ-summability and Brownian motion.

Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence

“…We now want to obtain some examples of convergence of interesting closed subgroups of quantum isometries in the setting of inductive limit of C*-algebras. In [12], the two last authors together with J. Packer proved a necessary and sufficient condition for the convergence, in the propinquity, of inductive limits of quantum compact metric spaces in terms of certain * -automorphisms called bridge builders, defined as follows. Definition 3.10.…”

“…With the notation of Definition (3.10), it was shown in [12,Theorem 2.22] that if we assume the existence of some M > 0 such that for all n ∈ N, we have 1 M L n L ∞ ML n over dom(L n ), then the existence of a bridge builder is equivalent to the convergence, in the sense of the propinquity, of (A n , L n ) n∈N to (A ∞ , L ∞ ). Moreover, if one finds a bridge builder which is a full quantum isometry, and if the quantum metrics are induced by metric spectral triples, then [12, Theorem 3.17] proves that the spectral triples themselves converge, in the stronger sense of the spectral propinquity -we refer to [20,19] for the construction and properties of the spectral propinquity.…”

“…For instance, spectral triples over AF algebras were first introduced by Christensen-Ivan in [6]; see [7], [8], [18], [1], [3] (among many other references) for further investigations of these spectral triples and generalizations. In a quite different direction, spectral triples on inductive limits of twisted group C*-algebras are introduced in [12] by the last two authors, together with J. Packer, with examples including the Bunce-Deddens algebras and the noncommutative solenoids (inductive limits of quantum tori) -C * -algebras which are not AF. These constructions were inspired by the work in [14] and [13].…”

“…(3) For a metric spectral triple, (A, H , / D) the subgroup Iso(A,H , / D) defined by: By [12,Theorem 2.22], given an inductive sequence of quantum compact metric spaces, whose limit is a quantum compact metric space, we have that the existence of a special type of * -automorphism on the inductive limit which is well behaved with respect to the quantum metrics, called a bridge builder, is equivalent, under a very natural assumption, to the convergence of the given sequence of quantum compact metric spaces for the propinquity. Examples of such construction were noncommutative solenoids and bounded Bunce-Deddens algebras, see [12]. Now, given that (metric) spectral triples were thus enriched with their isometry groups structures, a natural question to ask was if one could extend the original bridge builder construction of [12] in such a way it insured the convergence of the associated isometry groups.…”

“…Examples of such construction were noncommutative solenoids and bounded Bunce-Deddens algebras, see [12]. Now, given that (metric) spectral triples were thus enriched with their isometry groups structures, a natural question to ask was if one could extend the original bridge builder construction of [12] in such a way it insured the convergence of the associated isometry groups. This is what we are doing in this work.…”