 
Summary: NOTES ON PRODUCT SYSTEMS
WILLIAM ARVESON
Abstract. We summarize the basic properties of continuous tensor
product systems of Hilbert spaces and their role in noncommutative
dynamics. These are lecture notes, not intended for publication.
1. Concrete Product Systems
In these notes we assume the reader is familiar with the definition and
basic properties of E0semigroups, and has some familiarity with their role in
noncommutative dynamics. Our purpose here is to discuss product systems
and their significance in the classification problem.
Let = {t : t 0} be an E0semigroup acting on B(H), where as
always, H denotes a separable Hilbert space. The product system of gives
rise to a classifying structure for cocycle conjugacy, and is defined as follows.
For every t > 0 let E(t) be the following linear space of operators
E(t) = {T B(H) : t(X)T = TX, X B(H)}.
The first thing to notice is that there is a natural inner product ·, · on E(t)
that makes it into a Hilbert space. Indeed, if S, T E(t), then one finds
that for every X B(H), one has
T
SX = T
