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