Spectral Metric Spaces on Extensions of C*-Algebras

TL;DR

This paper develops a method to construct spectral triples on extensions of C*-algebras, enabling the creation of new quantum metric spaces and providing fresh insights into quantum spheres within noncommutative geometry.

Contribution

It introduces a novel construction of spectral triples on C*-algebra extensions that preserves summability and yields new quantum metric spaces, including on quantum spheres.

Findings

Constructed spectral triples on algebra extensions with Toeplitz property.

Produced new spectral triples on quantum 2- and 3-spheres.

Enhanced understanding of noncommutative geometric structures.

Abstract

We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.