Spectral triples and wavelets for higher-rank graphs

Abstract

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $\Lambda$, via the infinite path space ${\Lambda }^{\infty }$ of $\Lambda$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of ${\Lambda }^{\infty }$ 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 $\Lambda$. 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 $\zeta$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $\mu$, and show that $\mu$ is a rescaled version of the measure $M$ on ${\Lambda }^{\infty }$ 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 $L^2({\Lambda }^{\infty },M)$ which was constructed by Farsi et al.

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