E0–SEMIGROUPS FOR CONTINUOUS PRODUCT SYSTEMS: THE NONUNITAL CASE

Abstract. Let B be a σ–unital C∗–algebra. We show that every strongly con-tinuous E0–semigroup on the algebra of adjointable operators on a full Hilbert B–module E gives rise to a full continuous product system of correspondences over B. We show that every full continuous product system of correspondences over 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 hypothe-ses there is a one-to-one correspondence between E0–semigroups up to stable cocycle conjugacy and continuous product systems up to isomorphism. This generalizes the results for unital B to the σ–unital case