Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras

TL;DR

This paper computes quantum isometry groups of Cuntz--Krieger algebras with spectral triples from topological Markov chains, revealing new quantum groups with ergodic actions unlike classical isometries.

Contribution

It introduces a new family of compact quantum groups acting on Cuntz--Krieger algebras, combining features from quantum automorphism groups and easy quantum groups.

Findings

Quantum isometry groups are computed for these algebras.

Quantum groups act ergodically on the Cuntz algebra.

Construction of a quantum ergodic action on the Cantor space.

Abstract

We compute the quantum isometry groups of Cuntz--Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups. Contrary to the classical isometry groups, whose actions on the Cuntz--Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra. This also leads to the construction of a (genuinely quantum) ergodic action of a compact matrix quantum group on the Cantor space.