alpha-COMPLETELY POSITIVE MAPS ON LOCALLY C*-ALGEBRAS, KREIN MODULES AND RADON-NIKODYM THEOREM

Abstract

In this paper, we study alpha-completely positive maps between locally C*-algebras. As a generalization of a completely positive map, an a-completely positive map produces a Krein space with indefinite metric, which is useful for the study of massless or gauge fields. We construct a KSGNS type representation associated to an a-completely positive map of a locally C*-algebra on a Krein locally C*-module. Using this construction, we establish the Radon-Nikodym type theorem for alpha-completely positive maps on locally C*-algebras. As an application, we study an extremal problem in the partially ordered cone of a-completely positive maps on a locally C*-algebra.