# 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)
• Published : 2013.01.01

#### 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.

#### Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

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