Covariant representations on Krein -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.