Factorisation of equivariant spectral triples in unbounded KK-theory

Abstract

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\theta$-deformations. In particular we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

arXiv comment30 pages

References in corpus (3)

• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Index theory for locally compact noncommutative geometries
• Nonunital spectral triples and metric completeness in unbounded KK-theory

Cited by in corpus (6)

• The $K$-theoretic bulk-edge correspondence for topological insulators
• Nonunital spectral triples and metric completeness in unbounded KK-theory
• Generalized Connes-Chern characters in KK-theory with an application to weak invariants of topological insulators
• Lifting spectral triples to noncommutative principal bundles
• Application of semifinite index theory to weak topological phases
• Factorization of Dirac operators on toric noncommutative manifolds