Covariant representations on Krein C*-modules associated to pairs of two maps

Abstract

In this paper we construct a KSGNS type covariant representation on a Krein C*-module for a covariant alpha-completely positive map, and the result is applied to construct a KSGNS type covariant representation associated with a pair of two maps (rho, Phi) where rho is a covariant alpha-completely positive map on a C*-algebra and Phi, is a covariant rho-map on a Krein C*-module. The KSGNS type covariant representation for a pair (rho, Phi) is applied to give a new covariant J-representation of a crossed product of a C*-algebra and a new covariant map of a crossed product of a Hilbert C*-module by a discrete group.