Abstract: Path spaces, continuous tensor products
and E_0-semigroups
\abstract
We classify all continuous
tensor product systems of Hilbert spaces which are
``infinitely divisible" in the sense that they have
an associated logarithmic structure. These results
are applied to the theory of \esg s to deduce that every
\esg\ possessing sufficiently many ``decomposable" operators
must be cocycle conjugate to a $CCR$ flow.
A {\it path space} is an abstraction of the set
of paths in a topological space, on which there is given
an associative rule of concatenation. A {\it metric path space}
is a pair $(P,g)$ consisting of a path space $P$ and a function
$g:P^2\to \Bbb C$ which behaves as if it were
the logarithm of a multiplicative
inner product.
The logarithmic structures associated with
infinitely divisible product systems are such objects.
The preceding results are based on a classification
of metric path spaces.
\endabstract