Contractive spectral triples for crossed products

TL;DR

This paper extends the theory of spectral triples in noncommutative geometry by introducing contractive spectral triples for crossed products with general discrete group actions, simplifying the structure and broadening applications.

Contribution

It develops a generalized framework for spectral triples involving group actions, replacing isometric conditions with contractive ones, and extends existing theories to more general discrete groups.

Findings

Introduces coaction spectral triples with metric notions.

Replaces isometric conditions with contractive conditions.

Broadens the applicability of spectral triples to general discrete groups.

Abstract

Connes showed that spectral triples encode (noncommutative) metric information. Further, Connes and Moscovici in their metric bundle construction showed that, as with the Takesaki duality theorem, forming a crossed product spectral triple can substantially simplify the structure. In a recent paper, Bellissard, Marcolli and Reihani (among other things) studied in depth metric notions for spectral triples and crossed product spectral triples for $Z$-actions, with applications in number theory and coding theory. In the work of Connes and Moscovici, crossed products involving groups of diffeomorphisms and even of \'{e}tale groupoids are required. With this motivation, the present paper develops part of the Bellissard-Marcolli-Reihani theory for a general discrete group action, and in particular, introduces coaction spectral triples and their associated metric notions. The isometric condition is replaced by the contractive condition.