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http://dx.doi.org/10.4134/JKMS.2013.50.1.061

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

Heo, Jaeseong (Department of Mathematics Research Institute for Natural Sciences Hanyang University)
Ji, Un Cig (Department of Mathematics Research Institute of Mathematical Finance Chungbuk National University)
Kim, Young Yi (Department of Mathematics Chungbuk National University)
Publication Information
Journal of the Korean Mathematical Society / v.50, no.1, 2013 , pp. 61-80 More about this Journal
Abstract
In this paper, we study ${\alpha}$-completely positive maps between locally $C^*$-algebras. As a generalization of a completely positive map, an ${\alpha}$-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 ${\alpha}$-completely positive map of a locally $C^*$-algebra on a Krein locally $C^*$-module. Using this construction, we establish the Radon-Nikod$\acute{y}$m type theorem for ${\alpha}$-completely positive maps on locally $C^*$-algebras. As an application, we study an extremal problem in the partially ordered cone of ${\alpha}$-completely positive maps on a locally $C^*$-algebra.
Keywords
locally $C^*$-algebra; Hilbert locally $C^*$-module; ${\alpha}$-completely positive map; J-representation; Krein module; minimal Krein quadruple; non-commutative Radon-Nikod$\acute{y}$m theorem;
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