Spectral triples and wavelets for higher-rank graphs
- Authors
- Farsi, Carla; Gillaspy, Elizabeth; Julien, Antoine; Kang, Sooran; Packer, Judith
- Issue Date
- 15-Feb-2020
- Publisher
- Academic Press Inc.
- Keywords
- Dixmier trace; Finitely summable spectral triple; Higher-rank graph; Laplace-Beltrami operator; Wavelets; ζ-function
- Citation
- Journal of Mathematical Analysis and Applications, v.482, no.2
- Journal Title
- Journal of Mathematical Analysis and Applications
- Volume
- 482
- Number
- 2
- URI
- https://scholarworks.bwise.kr/cau/handle/2019.sw.cau/38181
- DOI
- 10.1016/j.jmaa.2019.123572
- ISSN
- 0022-247X
1096-0813
- 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(Λ∞,M) which was constructed by Farsi et al. © 2019
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