Motivated by the notion of P-functional, we introduce a notion of α-completely positive map between  ∗-algebras which is a Hermitian map satisfying a certain positivity condition, and then a α-completely positive map which is not completely positive is constructed. We establish the Kasparov-Stinespring-Gelfand-Naimark-Segal constructions of C∗-algebra and  ∗-algebra on Krein C∗-modules with α-completely positive maps.

In this paper we discuss reproducing kernels whose ranges are contained in a C∗-algebra or a Hilbert C∗-module. Using the construction of a reproducing Hilbert C∗-module associated with a reproducing kernel, we show how such a reproducing kernel can naturally be expressed in terms of operators on a Hilbert C∗-module using representations on Hilbert C∗-modules. We prove that for each positive definite G-kernel associated with a cocycle there is a representation associated with an operator-valued cocycle on the corresponding Hilbert C∗-module. Finally, some examples will be considered.

Link to this article: http://journals.cambridge.org/abstract_S0305004101005503How to cite this article: JAESEONG HEO (2002). A Radon-Nikodým theorem for completely positive invariant multilinear maps and its applications.

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We introduce a new notion of α-completely positive map on a C∗-algebra as a generalization of the notion of completely positive map. Then we study a theorem of the Radon–Nikodým type that there is a one-to-one correspondence between α-completely positive maps and positive operators and, as an application of the Radon–Nikodým type theorem, we give a characterization of pure α-completely positive maps. Finally, we study a covariant version of the Stinespring’s theorem for a covariant α-completely positive map (see Theorem 4.3).