Nondegenerate Representations of Continuous Product Systems

TL;DR

This paper proves that all continuous faithful product systems can be represented nondegenerately and continuously, extending known results from Hilbert spaces to Hilbert modules, with applications to C*-algebras and CP-semigroups.

Contribution

It establishes the existence of continuous faithful nondegenerate representations for all continuous faithful product systems, generalizing Arveson's results beyond Hilbert spaces.

Findings

Every continuous faithful product system admits a continuous faithful nondegenerate representation.

Extension of Arveson's result from Hilbert spaces to Hilbert modules.

New results on elementary dilations for (semi-)faithful CP-semigroups.

Abstract

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.