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

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.

v3 adds a discussion of the notion of quantum isometries used (after Definition 3.4) and a new reference, 35 pages The final version of the paper will appear in Communications in Mathematical Physics