2018/03/25 by Carla Farsi, Farsi, Carla, Elizabeth Gillaspy +7
Mathematics · Physics and Astronomy · #46L05 #46L87 #58J42 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #math-ph #math.FA #math.MP #math.OA #msc:46L05 #msc:46L87 #msc:58J42
paper · pdf · doi:10.48550/arxiv.1803.09304
This paper is a partial replacement of arXiv:1701.05321; the latter will not be submitted for publication. v2: Section 3 has been extensively revised to include a more detailed treatment of Dixmier traces. This version to appear in J. Math. Anal. Appl
arxiv created 2019/10/04 · arxiv updated 2019/10/07
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 k-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 ζ-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure μ, and show that μ is a rescaled version of the measure M 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 L2(Λ^∞, M) which was constructed by Farsi et al.