As a generalization of covariant completely positive maps, we consider (projective) covariant α-completely positive maps between locally C * -algebras. We first study (projective) covariant J-representations of locally C * -algebras on Krein modules over locally C * -algebras. Secondly, we construct a covariant KS-GNS type representation associated with a covariant α-completely positive map on a locally C * -algebra, and then study extensions to a locally C * -crossed product of α-completely positive maps on a locally C * -algebra. The results provide (projective) covariant representations of a locally C * -crossed product on a Krein module over a locally C * -algebra.
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.
In this paper we study unitary operator-valued multiplier σ on a normal subsemigroup S of a group G with its extension to G. A dilation of a projective isometric σ -representations of S to a projective unitary Φ(σ )-representation of G is established for a suitable unitary operator-valued multiplier Φ(σ ) associated with the multiplier σ which is explicitly constructed during the study.
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