On isometries of spectral triples associated to $AF$-algebras and crossed products

TL;DR

This paper investigates the structure of isometry groups for spectral triples on AF-algebras and crossed products, providing complete characterizations and demonstrating their coincidence in specific cases like the Cantor set.

Contribution

It fully determines the isometry group for spectral triples on AF-algebras and shows the applicability of crossed product spectral triples for lifting isometries.

Findings

Complete determination of the isometry group for AF-algebra spectral triples.

Coincidence of isometry groups in the case of the Cantor set.

Spectral triples on crossed products can be used to lift isometries.

Abstract

We examine the structure of two possible candidates of isometry groups for the spectral triples on $AF$-algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park, and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White and Zacharias, is suitable for the purpose of lifting isometries.