2017/01/19 by Carla Farsi, Farsi, Carla, Elizabeth Gillaspy +7 · 2 citations
Mathematics · #46L05 #46L87 #58J42 #Advanced Operator Algebra Research #FOS: Mathematics #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math.OA #msc:46L05 #msc:46L87 #msc:58J42
paper · pdf · doi:10.48550/arxiv.1701.05321
arxiv created 2017/01/19 · openalex publication_date 2017/01/19 · arxiv updated 2017/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we present two new ways to associate a spectral triple to a higher-rank graph Λ. Moreover, we prove that these 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. We first introduce the concept of stationary k-Bratteli diagrams, to associate a family of ultrametric Cantor sets to a finite, strongly connected higher-rank graph Λ. Then we show that under mild hypotheses, the Pearson-Bellissard spectral triples of such Cantor sets have a regular ζ-function, whose abscissa of convergence agrees with the Hausdorff dimension of the Cantor set, and that the measure μ induced by the associated Dixmier trace agrees with the measure M on the infinite path space Λ^∞ of Λ which was introduced by an Huef, Laca, Raeburn, and Sims. Furthermore, we prove that μ= M is a rescaled version of the Hausdorff measure of the ultrametric Cantor set. From work of Julien and Savinien, we know that for ζ-regular Pearson-Bellissard spectral triples, the eigenspaces of the associated Laplace-Beltrami operator constitute an orthogonal decomposition of L2(Λ^∞, μ); we show that this orthogonal decomposition refines the wavelet decomposition of Farsi et al. In addition, we generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph C^*-algebras, and prove that the wavelet decomposition of Farsi et al.~describes the eigenspaces of its Dirac operator.