Strict Completely Positive Maps between Locally C ^* -Algebras and Representations on Hilbert Modules

Abstract

The structure of the continuous strict completely positive linear maps between locally C * -algebras is described.

Completely multi-positive linear maps between locally C*-algebras and representations on Hilbert modules

“…) which satisfy conditions (a)-(c) of (1) and there is a unitary operator U q in L B q (E [̺ q ], F q ) which satisfies conditions (i)-(iii) of (2). Moreover,…”

“…The proofs are not always straightforward. In [2], the author extended the KSGNS (Kasparov, Stinespring, Gel'fand, Naimark, Segal) construction to a strict continuous, completely positive linear map from a locally C * -algebra A to L B (E), the locally C * -algebra of all adjointable operators on a Hilbert module E over a locally C * -algebra B. A covariant version of this construction is proved in [5].…”

Tensor Products of Hilbert Modules Over Locally C *-Algebras

“…They were first considered independently by A. Mallios in [7] and N. C. Phillips in [8], where the latter showed that most of the basic properties of Hilbert C * -modules are valid for Hilbert modules over locally C * -algebras. The Hilbert modules over locally C * -algebras are also studied in [4], [5] and elsewhere. Thus in [4] the present author proved a stabilization theorem for countably generated Hilbert modules over locally C * -algebras and in [5] she proved a version of the classical KSGNS (Kasparov, Stinespring, Gel'fand, Segal, Naimark) construction in the context of Hilbert modules over locally C * -algebras.…”

“…The Hilbert modules over locally C * -algebras are also studied in [4], [5] and elsewhere. Thus in [4] the present author proved a stabilization theorem for countably generated Hilbert modules over locally C * -algebras and in [5] she proved a version of the classical KSGNS (Kasparov, Stinespring, Gel'fand, Segal, Naimark) construction in the context of Hilbert modules over locally C * -algebras.…”

α-COMPLETELY POSITIVE MAPS ON LOCALLY C*-ALGEBRAS, KREIN MODULES AND RADON-NIKODÝM THEOREM

“…Bhatt and Karia [2] gave the Stinespring's construction for locally C * -algebras and Joiţa [10] generalized the KSGNS construction in the context of Hilbert modules over locally C * -algebras. Let p be any element in S(A).…”