E${}_0$-semigroups and product systems of W${}^{\ast }$-bimodules

Abstract

Product systems have been originally introduced to classify E${}_0$-semigroups on type I factors by Arveson. We develop the classification theory of E${}_0$-semigroups on a general von Neumann algebra and the dilation theory of CP${}_0$-semigroups in terms of W${}^{\ast }$-bimodules. For this, we provide a notion of product system of W${}^{\ast }$-bimodules. This is a W${}^{\ast }$-bimodule version of Arveson's and Bhat-Skeide's product systems. There exists a one-to-one correspondence between CP${}_0$-semigroups and units of product systems of W${}^{\ast }$-bimodules. The correspondence implies a construction of a dilation of a given CP${}_0$-semigroup, a classification of E${}_0$-semigroups on a von Neumann algebra up to cocycle equivalence and a relationship between Bhat-Skeide's and Muhly-Solel's constructions of minimal dilations of CP${}_0$-semigroups.

22 pages