Spectral triples for higher-rank graph $C^*$-algebras

TL;DR

This paper introduces a new spectral triple for higher-rank graph $C^*$-algebras, linking it to wavelet decompositions of their infinite path spaces, extending prior work on Cuntz-Krieger algebras.

Contribution

It generalizes spectral triples from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras and connects them to wavelet decompositions of infinite path spaces.

Findings

Spectral triples are constructed for higher-rank graph $C^*$-algebras.

The eigenspaces of the Dirac operator correspond to wavelet decomposition.

The approach extends previous spectral triples to a broader class of algebras.

Abstract

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(\Lambda)$ of a strongly connected finite higher-rank graph $\Lambda$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(\Lambda)$, and we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $\Lambda$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of this spectral triple.