A Radon-Nikodym type theorem for $\alpha$-completely positive maps on groups

TL;DR

This paper establishes a Radon-Nikodym type theorem for $oldsymbol{ ext{alpha}}$-completely positive maps on groups, linking these maps to unitary representations on Krein spaces and exploring their equivalence and ordering relations.

Contribution

It introduces a Radon-Nikodym type theorem for $oldsymbol{ ext{alpha}}$-completely positive maps, characterizing them via unitary representations on Krein spaces and defining a pre-order relation.

Findings

Operator valued $oldsymbol{ ext{alpha}}$-completely positive maps correspond to unitary representations on Krein spaces.

Unitary equivalence of representations implies the same $oldsymbol{ ext{alpha}}$-completely positive map.

A pre-order relation on these maps is characterized through their associated unitary representations.

Abstract

We show that an operator valued $\alpha$-completely positive map on a group G is given by a unitary representation of G on a Krein space which satisfies some condition. Moreover, two unitary equivalent such unitary representations define the same {\alpha}-completely positive map. Also we introduce a pre-order relation on the collection of {\alpha}-completely positive maps on a group and we characterize this relation in terms of the unitary representation associated to each map.