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

The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert C∗C∗-module is isomorphic to a direct summand in the standard module of square summable sequences in the base C∗C∗-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base C∗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 correpondences) on Hilbert C∗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∗C∗-module.