Factorisation of equivariant spectral triples in unbounded KK-theory

2015/05/12 by Iain Forsyth, Forsyth, Iain, Adam Rennie +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1505.02863

openalex publication_date 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 θ-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).

Citations

• Index theory for locally compact noncommutative geometries

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• The G-signature Theorem on Witt spaces

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