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Journal of Operator Theory

Volume 65, Issue 1, Winter 2011  pp. 71-85.

Nondegenerate representations of continuous product systems

Authors:  Michael Skeide
Author institution: Dipartimento S.E.G.e S., Universita degli Studi del Molise, Via de Sanctis, 86100 Campobasso, Italy

Summary:  We show that every (continuous) faithful product system admits a (continuous) faithful nondegenerate representation. For Hilbert spaces this is equivalent to Arveson's result that every Arveson system comes from an $E_0$-semigroup. We point out that for Hilbert modules this is not so. As applications we show a $C^*$-algebra version of a result for von Neumann algebras due to Arveson and Kishimoto, and a result about existence of elementary dilations for (semi-)faithful CP-semigroups.

Keywords:  Quantum probability, quantum dynamics, product systems, Hilbert modules, representations

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