Let $\mathcal{B}$ 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 $\mathcal{B}$-module $E$ gives rise to a full continuous product system
of correspondences over $\mathcal{B}$. We show that every full continuous
product system of correspondences over $\mathcal{B}$ 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 $\mathcal{B}$ to the $\sigma$-unital case.
@article{1261086705,
author = {Skeide, Michael},
title = {$E\_0$--Semigroups for Continuous Product Systems: The Nonunital Case},
journal = {Banach J. Math. Anal.},
volume = {3},
number = {2},
year = {2009},
pages = { 16-27},
language = {en},
url = {http://dml.mathdoc.fr/item/1261086705}
}
Skeide, Michael. $E_0$--Semigroups for Continuous Product Systems: The Nonunital Case. Banach J. Math. Anal., Tome 3 (2009) no. 2, pp. 16-27. http://gdmltest.u-ga.fr/item/1261086705/