Journal of the
Korean Mathematical Society JKMS

In this paper, we study $\alpha$-completely positive maps between locally $C^*$-algebras. As a generalization of a completely positive map, an $\alpha$-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 $\alpha$-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 $\alpha$-completely positive maps on locally $C^*$-algebras. As an application, we study an extremal problem in the partially ordered cone of $\alpha$-completely positive maps on a locally $C^*$-algebra.

Keywords: locally $C^*$-algebra, Hilbert locally $C^*$-module, $\al$-completely positive map, $J$-representation, Krein module, minimal Krein quadruple, non-commutative Radon-Nikod\'{y}m theorem

MSC numbers: Primary 46L08, 46K10; Secondary 47L60, 46L55