Extensions and Dilations of module maps

TL;DR

This paper extends classical dilation and extension theorems for completely positive maps to the setting of $C^*$-modules, broadening their applicability in operator algebra theory.

Contribution

It generalizes key dilation and extension results, such as Stinespring and Wittstock theorems, to the context of $C^*$-module maps with compatible actions.

Findings

Extended Stinespring dilation theorem for $C^*$-module maps

Generalized Wittstock and Arveson extension theorems in the module setting

Broadened the theoretical framework for completely positive module maps

Abstract

We study completely positive module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. We extend several well known dilation and extension results to this setup, including the Stinespring dilation theorem and Wittstock, Arveson, and Voiculescu extension theorems.