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

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