A Connes correspondence approach to the dilation theory

Abstract

Product systems have been originally introduced to classify E0-semigroups on type I factors by Arveson. The idea of product systems is influenced the constructions of dilations of CP0-semigroups. In this paper, we will develop the dilation theory and the classification theory of E0-semigroups on a general von Neumann algebra in the view point of Connes correspondences. For this, we will provide a concept of product system whose components are Connes correspondences. There exists a one-to-one correspondence between a CP0-semigroup and a unit of a product system of Connes correspondences. The correspondence enables us to construct dilations and classify E0-semigroups by product systems of Connes correspondences up to cocycle equivalence.