On KSGNS representations on Krein C*-modules

Motivated by the notion of P-functional, we introduce a notion of alpha-completely positive map between *-algebras which is a Hermitian map satisfying a certain positivity condition, and then a alpha-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 alpha-completely positive maps. (C) 2010 American Institute of Physics. [doi:10.1063/1.3397448]J.H. was supported by Korea Research Foundation Grant funded by the Korean Government (Grant No. KRF-2008-314-C00014). U.C.J. was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (Grant No. R01-2008-000-10843-0)