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

In this paper we construct a KSGNS type covariant representation on a Krein C∗-module for a covariant α-completely positive map, and the result is applied to construct a KSGNS type covariant representation associated with a pair of two maps (ρ,Φ) where ρ is a covariant α-completely positive map on a C∗-algebra and Φ is a covariant ρ-map on a Krein C∗-module. The KSGNS type covariant representation for a pair (ρ,Φ) 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.