Covariant version of the Stinespring type theorem for Hilbert C*-modules

Abstract

We prove a covariant version of the Stinespring theorem for Hilbert C * -modules.2000 Mathematics Subject Classification. Primary 46L08.

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

“… ρ(f ) = V * (π × v)(f )V =  G V * π (f (s))v s Vds =  G ρ(f (s))u s ds, which gives the proof of (14). The conditions (i) and (ii) in Definition 2.1 are satisfied by the above argument.…”

“…Moreover, Heo-Ji [12] constructed a Stinespring type covariant representation for a pair of a covariant completely positive map ρ and a covariant ρ-map. In this paper, motivated by the results in [3,13,11,12,14], we construct a KSGNS type covariant representation for a pair of a covariant α-completely positive map ρ on a C * -algebra and a covariant ρ-map on a Krein C * -module, using the KSGNS type representations on Krein C * -modules associated to α-completely positive maps. Furthermore, we give a new covariant J-representation of a crossed product of a C * -algebra by a locally compact group and a new covariant map on the crossed product of a Hilbert C * -module by a locally compact group.…”

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]).…”

$\mathfrak K$-families and CPD-H-extendable families

“…[BRS12]). Certain related covariant versions of this theorem have been explored in [Joi11] and [Heo99].…”

“…If (G, η, E) is a dynamical system on a full Hilbert B-module E, then there exists unique C * -dynamical system (G, α η , B) (cf. p.806 of [Joi11]) such that…”