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

摘要

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.