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ON THE EXISTENCE OF E0-SEMIGROUPS WILLIAM ARVESON
 

Summary: ON THE EXISTENCE OF E0-SEMIGROUPS
WILLIAM ARVESON
Abstract. Product systems are the classifying structures for semi-
groups of endomorphisms of B(H), in that two E0-semigroups 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 E0-semigrouop. 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 E0-semigroups up to co-
cycle conjugacy, in that two E0-semigroups 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 E0-semigroup.
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 E0-semigroup 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

  

Source: Arveson, William - Department of Mathematics, University of California at Berkeley

 

Collections: Mathematics