COMPLETELY MULTI-POSITIVE LINEAR MAPS AND REPRESENTATIONS ON HILBERT C -MODULES

Abstract

We introduce the notion of (completely) multi-positive linear maps between C -algebras, and show that a completely multi-positive lin- ear map induces a representation of a C -algebra on Hilbert C -modules. This generalizes the Stinespring's representation and the representations con- structed by Paschke and Kaplan as well as the GNS representation. We also construct the covariant representations on Hilbert C -modules for covari- ant 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.