Minimal E_0-semigroups

TL;DR

This paper provides a new characterization of minimality for E_0-semigroups on von Neumann algebras, clarifying when such semigroups are uniquely determined up to conjugacy, with broad applicability.

Contribution

It introduces a novel characterization of minimality using projective cocycles and their limits, applicable to general von Neumann algebras with separable predual.

Findings

New criteria for minimality in terms of projective cocycles

Clarification of conditions for uniqueness of E_0-semigroups

Applicability to arbitrary von Neumann algebras with separable predual

Abstract

It is known that every semigroup of normal completely positive maps of a von Neumann can be ``dilated" in a particular way to an E_0-semigroup acting on a larger von Neumann algebra. The E_0-semigroup is not uniquely determined by the completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of {\it minimality} are met. Minimality is a subtle property, and it is often not obvious if it is satisfied for specific examples even in the simplest case where the von Neumann algebra is $\Cal B(H)$. In this paper we clarify these issues by giving a new characterization of minimality in terms projective cocycles and their limits. Our results are valid for semigroups of endomorphisms acting on arbitrary von Neumann algebras with separable predual.