KSGNS Type Constructions for $$\alpha $$ α -Completely Positive Maps on Krein $$C^*$$ C ∗ -Modules

Abstract

In this paper, we investigate Φ-maps associated to a certain type of αcompletely positive maps. We then prove a KSGNS (Kasparov-Stinespring-Gel'fand-Naimark-Segal) type theorem for α-completely positive maps on Krein C * -modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant α-completely positive map and study the structure of minimal covariant KSGNS constructions.1991 Mathematics Subject Classification. Primary 46L08; Secondary 47B50, 46L05.

Matrix KSGNS construction and a Radon–Nikodym type theorem

“…A completely positive map ϕ : A → B of locally C * -algebras is a linear map such that ϕ n : M n (A ) → M n (B) defined by ϕ n (a ij )) n i,j=1 = (ϕ(a ij )) n i,j=1 is positive. Stinespring [18] showed that a completely positive linear map ϕ from A to the C * -algebra L (H) of all bounded linear operators acting on a Hilbert space H is of the form ϕ(·) = S * π(·)S, where π is a * -representation of A on a Hilbert space K and S is a bounded linear operator from H to K. Nowadays, the theory of completely positive linear maps on Hilbert and Krein A-modules is a vast area of the modern analysis (see [3,4,5,11,12,13,14,16]).…”