E_0-Semigroups for Continuous Poduct Systems: The Nonunital Case

TL;DR

This paper extends the theory of E_0-semigroups and continuous product systems of correspondences to sigma-unital C*-algebras, establishing a one-to-one correspondence under countability conditions.

Contribution

It generalizes the classification of E_0-semigroups and product systems from unital to sigma-unital C*-algebras, including countability considerations.

Findings

Every strongly continuous E_0-semigroup induces a full continuous product system.

Every full continuous product system arises from an E_0-semigroup.

A one-to-one correspondence exists between E_0-semigroups and product systems under countability conditions.

Abstract

Let 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 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 countable 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-semigroup up to stable cocycle conjugacy and continuous product systems up isomorphism. This generalizes the results for unital B to the sigma-unital case.