Sums of regular selfadjoint operators in Hilbert-C*-modules

TL;DR

This paper introduces a new notion of weak anticommutativity for self-adjoint operators in Hilbert-C*-modules, proving the self-adjointness of their sums and connecting it to KK-theory, thus weakening existing conditions for Kasparov product representation.

Contribution

It defines weak anticommutativity for self-adjoint operators in Hilbert-C*-modules and proves sum self-adjointness, linking to KK-theory and weakening Kucerovsky's conditions.

Findings

Sum of weakly anticommuting operators is self-adjoint and regular.

The new notion relates to Connes-Skandalis positivity in KK-theory.

Conditions are close to optimal for representing Kasparov products.

Abstract

We introduce a notion of weak anticommutativity for a pair (S,T) of self-adjoint regular operators in a Hilbert-C*-module E. We prove that the sum $S+T$ of such pairs is self-adjoint and regular on the intersection of their domains. A similar result then holds for the sum S^2+T^2 of the squares. We show that our definition is closely related to the Connes-Skandalis positivity criterion in $KK$-theory. As such we weaken a sufficient condition of Kucerovsky for representing the Kasparov product. Our proofs indicate that our conditions are close to optimal.