Path spaces, continuous tensor products, and E_0 semigroups

TL;DR

This paper classifies continuous tensor product systems of Hilbert spaces with an infinite divisibility property and applies these results to classify certain E_0 semigroups, revealing their structure via path space abstractions.

Contribution

It introduces a classification of infinitely divisible continuous tensor product systems and applies this to characterize E_0 semigroups with decomposable operators as CCR flows.

Findings

Classified all infinitely divisible continuous tensor product systems.

Proved that E_0 semigroups with enough decomposable operators are cocycle conjugate to CCR flows.

Developed a classification of metric path spaces related to these structures.

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 E_0 semigroups to deduce that every E_0 semigroup which possesses 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 --> complex numbers 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.