Factorisation of equivariant spectral triples in unbounded KK-theory

Understand

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).