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Zbl 1193.46043
Skeide, Michael
$E_0$-semigroups for continuous product systems: the nonunital case. (English)
[J] Banach J. Math. Anal. 3, No. 2, 16-27, electronic only (2009). ISSN 1735-8787/e

Summary: Let $\cal B$ be a $\sigma$-unital $C^*$-algebra. We show that every strongly continuous $E_0$-semigroup on the algebra of adjointable operators on a full Hilbert $\cal B$-module $E$ gives rise to a full continuous product system of correspondences over $\cal B$. We show that every full continuous product system of correspondences over $\cal B$ arises in that way. If the product system is countably generated, then $E$ can be chosen countably generated, and if $E$ is countably generated, then so is the product system. We show that under these countability hypotheses there is a one-to-one correspondence between $E_0$-semigroups up to stable cocycle conjugacy and continuous product systems up to isomorphism. This generalizes the results for unital $\cal B$ to the $\sigma $-unital case.

MSC 2000:: *46L55 Noncommutative dynamical systems
46L53 Noncommutative probability and statistics
46L08 C*-modules

Keywords: quantum probability; quantum dynamics; product system; Hilbert module

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