 
Summary: ON THE EXISTENCE OF E0SEMIGROUPS
WILLIAM ARVESON
Abstract. Product systems are the classifying structures for semi
groups of endomorphisms of B(H), in that two E0semigroups are co
cycle conjugate iff their product systems are isomorphic. Thus it is im
portant to know that every abstract product system is associated with
an E0semigrouop. This was first proved more than fifteen years ago by
rather indirect methods. Recently, Skeide has given a more direct proof.
In this note we give yet another proof by a very simple construction.
1. Introduction, formulation of results
Product systems are the structures that classify E0semigroups up to co
cycle conjugacy, in that two E0semigroups are cocycle conjugate iff their
concrete product systems are isomorphic [Arv89]. Thus it is important to
know that every abstract product system is associated with an E0semigroup.
There were two proofs of that fact [Arv90], [Lie03] (also see [Arv03]), both
of which involved substantial analysis. In a recent paper, Michael Skeide
[Ske06] gave a more direct proof. In this note we present a new and simpler
method for constructing an E0semigroup from a product system.
Our terminology follows the monograph [Arv03]. Let E = {E(t) : t > 0}
be a product system and choose a unit vector e E(1). e will be fixed
