$alpha$-completely positive maps on locally $C^*$-algebras, Krein modules and Radon-Nikod'ym theorem

摘要

In this paper, we study $lpha$-completely positive maps between locally $C^*$-algebras. As a generalization of a completely positive map, an $lpha$-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 $lpha$-completely positive map of a locally $C^*$-algebra on a Krein locally $C^*$-module. Using this construction, we establish the Radon-Nikod'ym type theorem for $lpha$-completely positive maps on locally $C^*$-algebras. As an application, we study an extremal problem in the partially ordered cone of $lpha$-completely positive maps on a locally $C^*$-algebra.