On spectral triples on crossed products arising from equicontinuous actions

TL;DR

This paper constructs spectral triples on crossed product $C^*$-algebras using the Kasparov product, providing new insights into their structure and applications to specific algebras like Bunce-Deddens and AF-algebras.

Contribution

It introduces a novel method to build spectral triples on crossed products via the Kasparov product, extending previous work and applying it to new classes of algebras.

Findings

Spectral triples constructed for crossed products by equicontinuous actions.

Application to Bunce-Deddens algebra from odometer actions.

Extension of spectral triples to AF-algebras and other crossed products.

Abstract

The external Kasparov product is used to construct odd and even spectral triples on crossed products of $C^*$-algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is $\Z$ this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.