The structure of invariant α-CP multilinear maps and associated J-representations

Abstract

In this paper, we introduce a notion of alpha-completely positive multilinear maps as a generalization of completely positive multilinear maps. We construct a Krein space representation associated with an invariant alpha-completely positive multilinear map and show that the natural ordering of invariant alpha-completely positive multilinear maps is characterized in terms of the Radon-Nikodym derivatives. Finally, we construct a covariant Krein space representation associated with covariant and invariant alpha-completely positive multilinear maps and show that a covariant and invariant alpha-completely positive multilinear map on C*-algebras can be extended to an alpha-completely positive multilinear map on C*-crossed products.