Completely semi-$\varphi$-maps

TL;DR

This paper introduces a new class of maps called completely semi-$$-maps on Hilbert $C^*$-modules, generalizing existing $$-maps, and develops a Stinespring-type representation and Radon-Nikodym theorem for them.

Contribution

It defines completely semi-$$-maps, provides a representation theorem similar to Stinespring's, and establishes a Radon-Nikodym theorem for this new class.

Findings

Provides examples of CP-extendable maps not CP-H-extendable.

Establishes a Stinespring-like representation theorem for completely semi-$$-maps.

Proves a Radon-Nikodym type theorem for these maps.

Abstract

We introduce completely semi-$\varphi$-maps on Hilbert $C^*$-modules as a generalization of $\varphi$-maps. This class of maps provides examples of CP-extendable maps which are not CP-H-extendable, in Skeide-Sumesh's sense. Using the CP-extendability of completely semi-$\varphi$-maps, we give a representation theorem, similar to Stinespring's representation theorem, for this class of maps which can be considered as strengthened and generalized form of Asadi's and Bhat-Ramesh-Sumesh's analogues of Stinespring representation theorem for $\varphi$-maps. We also define an order relation on the set of all completely semi-$\varphi$-maps and establish a Radon-Nikodym type theorem for this class of maps in terms of their representations.