Differentiable absorption of Hilbert C*-modules, connections, and lifts of unbounded operators

TL;DR

This paper extends the Kasparov absorption theorem to a differentiable setting using derivations on C*-algebras, enabling the construction of connections and lifts of unbounded operators on Hilbert modules.

Contribution

It introduces a differentiable version of the Kasparov absorption theorem incorporating derivations, and applies it to construct connections and lifts of unbounded operators.

Findings

Established a differentiable absorption theorem with minimal assumptions

Constructed densely defined connections on Hilbert C*-modules

Defined selfadjoint and regular lifts of unbounded operators

Abstract

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C*-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C*-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C*-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correspondences) on Hilbert C*-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert C*-module.