an: Zbl 1193.46043
au: Skeide, Michael
ti: $E_0$-semigroups for continuous product systems: the nonunital case.
la: EN
so: Banach J. Math. Anal. 3, No. 2, 16-27, electronic only (2009).
py: 2009
dt: J
cc: *46L55 46L53 46L08
ut: quantum probability; quantum dynamics; product system; Hilbert module
ab: 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.