Path Spaces , Continuous Tensor Products

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 E0-semigroups to deduce that every E0-semigroup possessing sufficiently many “decomposable” operators must be cocycle conjugate to a CCR flow. A 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 metric path space is a pair (P, g) consisting of a path space P and a function g : P 2 → 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.