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

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