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

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.