Heat operators and isometry groups of Cuntz-Krieger algebras

TL;DR

This paper develops heat semigroups and spectral triples for Cuntz-Krieger algebras using spectral noncommutative geometry, linking their structure to graph automorphisms and entropy properties.

Contribution

It introduces a novel construction of spectral triples on Cuntz-Krieger algebras and explicitly computes heat operators for Cuntz algebras, connecting geometric and algebraic features.

Findings

Spectral triples exhaust odd K-homology of Cuntz-Krieger algebras.

Heat operators for Cuntz algebras are explicitly computed as Riesz potentials.

Isometry groups correspond to automorphisms of the underlying graph, with vanishing Voiculescu entropy.

Abstract

This paper introduces heat semigroups of topological Markov chains and Cuntz-Krieger algebras by means of spectral noncommutative geometry. Using recent advances on the logarithmic Dirichlet Laplacian on Ahlfors regular metric-measure spaces, we construct spectral triples on Cuntz-Krieger algebras from singular integral operators. These spectral triples exhaust odd K-homology and for Cuntz algebras we can compute their heat operators explicitly as Riesz potential operators. We also describe their isometry group in terms of the automorphism group of the underlying directed graph and prove that the Voiculescu noncommutative topological entropy vanishes on isometries.