alpha-completely positive maps of group systems and Krein module representations

Abstract

In this paper, we study (covariant) alpha-completely positive maps on group systems. We first introduce a notion of alpha-completely positive maps of groups into (locally) C*-algebras and show that bounded alpha-completely positive maps on discrete groups induce alpha-completely positive linear maps on group C*-algebras. We establish the (covariant) KSGNS type representation theorem for (covariant) alpha-completely positive maps of group systems into locally C*-algebras. These constructions provide a projective covariant J-representation of a group system into a locally C*-algebra.