Journal of Operator Theory

Volume 41, Issue 1, Winter 1999 pp. 3-22.

Completely multi-positive linear maps and representations on Hilbert

$C^*$ -modules

Authors: Jaeseong Heo
Author institution: Department of Mathematics, Seoul National University, Seoul 151-742, Korea

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.

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