Differentiable absorption of Hilbert $C^{\ast }$-modules, connections, and lifts of unbounded operators

Abstract

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert $C\ast$-module is isomorphic to a direct summand in the standard module of square summable sequences in the base $C\ast$-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base $C\ast$-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 correpondences) on Hilbert $C\ast$-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert $C\ast$-module.