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FACTORISATION OF EQUIVARIANT SPECTRAL TRIPLES IN UNBOUNDED $KK$-THEORY

Part of: Selfadjoint operator algebras $K$-theory and operator algebras

Published online by Cambridge University Press:  21 December 2018

IAIN FORSYTH  and
ADAM RENNIE
IAIN FORSYTH
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, Australia School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia Institut für Analysis, Leibniz Universität Hannover, Germany email iainforsyth@hotmail.com
ADAM RENNIE*
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, Australia email renniea@uow.edu.au
*
✉renniea@uow.edu.au
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Abstract

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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. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$-deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. 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).

Keywords

KK-theoryspectral tripleKasparov product

MSC classification

Primary: 19K35: Kasparov theory ($KK$-theory)
Secondary: 46L87: Noncommutative differential geometry
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 2 , October 2019 , pp. 145 - 180
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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