Extensions and degenerations of spectral triples

TL;DR

This paper constructs a family of spectral triples for unital C*-algebras with extensions, analyzing their metric properties and limits, and applies these to the unitarized compact operators and Podles sphere.

Contribution

It introduces a parameterized family of spectral triples for extensions of C*-algebras and studies their metric continuity and limiting behaviors.

Findings

Spectral triples vary continuously with parameters (s,t).

Limits of spectral triples recover known geometries like unitarized compacts.

The constructed triples for Podles sphere satisfy key axioms of noncommutative geometry.

Abstract

For a unital C*-algebra A, which is equipped with a spectral triple and an extension T of A by the compacts, we construct a family of spectral triples associated to T and depending on the two positive parameters (s,t). Using Rieffel's notation of quantum Gromov-Hausdorff distance between compact quantum metric spaces it is possible to define a metric on this family of spectral triples, and we show that the distance between a pair of spectral triples varies continuously with respect to the parameters. It turns out that a spectral triple associated to the unitarization of the algebra of compact operators is obtained under the limit - in this metric - for (s,1) -> (0, 1), while the basic spectral triple, associated to A, is obtained from this family under a sort of a dual limiting process for (1, t) -> (1, 0). We show that our constructions will provide families of spectral triples for the unitarized compacts and for the Podles sphere. In the case of the compacts we investigate to which extent our proposed spectral triple satisfies Connes' 7 axioms for noncommutative geometry.