Spectral triples for noncommutative solenoidal spaces from self-coverings

TL;DR

This paper constructs spectral triples on noncommutative solenoidal spaces derived from self-coverings, showing their convergence to a semifinite spectral triple and analyzing their metric properties.

Contribution

It extends spectral triples from base spaces to inductive families of coverings, introducing noncommutative solenoidal spaces with specific metric and volume characteristics.

Findings

Spectral triples on coverings converge to a semifinite spectral triple.

Noncommutative solenoidal spaces share metric dimension and volume with base spaces.

These spaces are not quantum compact metric spaces, as their pseudo-metric does not induce the weak* topology.

Abstract

Examples of noncommutative self-coverings are described, and spectral triples on the base space are extended to spectral triples on the inductive family of coverings, in such a way that the covering projections are locally isometric. Such triples are shown to converge, in a suitable sense, to a semifinite spectral triple on the direct limit of the tower of coverings, which we call noncommutative solenoidal space. Some of the self-coverings described here are given by the inclusion of the fixed point algebra in a C$^*$-algebra acted upon by a finite abelian group. In all the examples treated here, the noncommutative solenoidal spaces have the same metric dimension and volume as on the base space, but are not quantum compact metric spaces, namely the pseudo-metric induced by the spectral triple does not produce the weak$^*$ topology on the state space.