Spectral triples and wavelets for higher-rank graphs

Carla Farsi, Elizabeth Gillaspy, Antoine Julien, Sooran Kang, Judith Packer

Research output: Contribution to journalArticlepeer-review

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 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,M) which was constructed by Farsi et al.

Original languageEnglish
Article number123572
JournalJournal of Mathematical Analysis and Applications
Volume482
Issue number2
DOIs
StatePublished - Feb 15 2020

Keywords

  • Dixmier trace
  • Finitely summable spectral triple
  • Higher-rank graph
  • Laplace-Beltrami operator
  • Wavelets
  • ζ-function

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