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

Abstract

Abstract. In this paper, we study α-completely positive maps between locally C * -algebras. As a generalization of a completely positive map, an α-completely positive map produces a Krein space with indefinite metric, which is useful for the study of massless or gauge fields. We construct a KSGNS type representation associated to an α-completely positive map of a locally C * -algebra on a Krein locally C * -module. Using this construction, we establish the Radon-Nikodým type theorem for α-completely positive maps on locally C * -algebras. As an application, we study an extremal problem in the partially ordered cone of α-completely positive maps on a locally C * -algebra.

Projective Covariant Representations of Locally $C^*$-Dynamical Systems

“…where F ρp is the completion of A qp ⊗ F p / ker( •, • p ) (see the proof of Theorem 3.1 in [7]). Hence we may assume that F ρ is the completion of A p ⊗ F/ ker( •, • p ).…”

“…In Section 3, we study a covariant representation of a locally C * -algebra. We consider an α-completely positive map on a locally C * -algebra and construct a KSGNS type representation of a locally C * -algebra A associated with an α-completely positive map ρ (see [7]). The construction leads to a J ρ -representation of the locally C * -algebra A on a Krein A-module.…”

BURES DISTANCE FOR $\alpha$-COMPLETELY POSITIVE MAPS AND TRANSITION PROBABILITY BETWEEN $P$-FUNCTIONALS

“…From the motivation in local quantum field theory, we introduced the notion of α-completely positive (α-CP) maps between (locally) C * -algebras, which generalizes α-positivity [15] and P -functionals [17]. The α-complete positivity was studied in several papers [8,9,10,11,12,13].…”