Addendum to: Indefinite Kasparov modules and pseudo-Riemannian manifolds

TL;DR

This paper refines the theory of indefinite Kasparov modules, broadening their applicability to include Dirac operators on pseudo-Riemannian manifolds by weakening initial assumptions and leveraging new operator sum theorems.

Contribution

It weakens the assumptions on indefinite Kasparov modules and demonstrates their equivalence to pairs of Kasparov modules, including key examples like pseudo-Riemannian Dirac operators.

Findings

Weaker assumptions still allow indefinite Kasparov modules to model non-symmetric operators.

The new theorem ensures self-adjointness and regularity of sums of weakly anticommuting operators.

Main example of Dirac operator on pseudo-Riemannian manifolds is now included.

Abstract

We improve our previous results on indefinite Kasparov modules, which provide a generalisation of unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g. hyperbolic) operators. In particular, we can weaken the assumptions that are imposed on indefinite Kasparov modules. Using a new theorem by Lesch and Mesland on the self-adjointness and regularity of the sum of two weakly anticommuting operators, we show that we still have an equivalence between indefinite Kasparov modules and pairs of Kasparov modules. Importantly, the weakened version of indefinite Kasparov modules now includes the main motivating example of the Dirac operator on a pseudo-Riemannian manifold. The appendix contains a construction of an approximate identity for weakly commuting operators, which is due to Lesch and Mesland.