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Completely multi-positive linear maps and representations on Hilbert C * -modules. (English) Zbl 0994.46019
Summary: We introduce the notion of (completely) multi-positive linear maps between C * -algebras, and show that a completely multi-positive linear map induces a representation of a C * -algebra on Hilbert C * -modules. This generalizes the Stinespring’s representation and the representations constructed by Paschke and Kaplan as well as the GNS representation. We also construct the covariant representations on Hilbert C * -modules for covariant completely positive linear maps. Using representations of C * -algebras on Hilbert C * -modules associated with completely multi-positive linear maps we establish another approach about representations associated with completely bounded linear maps.

MSC:
46L08C * -modules
46L40Automorphisms of C * -algebras
46H25Normed modules and Banach modules, topological modules