Hilbert C∗-modules and projective representations associated with multipliers

In this paper we consider projective σ-representations and give the dilations associated with the representations. To unify the various representation theorem, we use the Kolmogorov decompositions for positive definite kernels. We present dilation theorems for projective σ-representations, which generalize Stinespring dilation theorem for covariant completely positive maps as well as Naimark–Sz.-Nagy characterization of positive definite functions. We also construct a representation of a right Hilbert C∗-module for a strictly U-covariant and completely positive map.