In this paper, we study

-completely positive maps between locally

-algebras. As a generalization of a completely positive map, an

-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

-completely positive map of a locally

-algebra on a Krein locally

-module. Using this construction, we establish the Radon-Nikod

m type theorem for

-completely positive maps on locally

-algebras. As an application, we study an extremal problem in the partially ordered cone of

-completely positive maps on a locally

-algebra.