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

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