# E$_0$-semigroups and product systems of W$^*$-bimodules

## TL;DR

This paper extends the classification and dilation theory of E$_0$-semigroups and CP$_0$-semigroups from type I factors to general von Neumann algebras using W$^*$-bimodules, introducing a new product system framework.

## Contribution

It develops a new theory of product systems of W$^*$-bimodules and establishes a correspondence with CP$_0$-semigroups, generalizing previous results to broader algebraic settings.

## Key findings

- Established a one-to-one correspondence between CP$_0$-semigroups and units of W$^*$-bimodule product systems
- Constructed dilations of CP$_0$-semigroups using the new product system framework
- Classified E$_0$-semigroups on von Neumann algebras up to cocycle equivalence

## 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$^*$-bimodules. For this, we provide a notion of product system of W$^*$-bimodules. This is a W$^*$-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$^*$-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.