Generalized CCR Flows

TL;DR

This paper introduces generalized CCR flows, a new class of $E_0$-semigroups, providing criteria for type III classification, constructing examples, and demonstrating the existence of uncountably many non-cocycle conjugate type III flows.

Contribution

It develops a new construction of $E_0$-semigroups called generalized CCR flows, with criteria for type III classification and examples that distinguish them from existing types.

Findings

New necessary and sufficient condition for type III classification.

Construction of uncountably many non-cocycle conjugate type III $E_0$-semigroups.

Examples indistinguishable from type I by previous invariants.

Abstract

We introduce a new construction of $E_0$-semigroups, called generalized CCR flows, with two kinds of descriptions: those arising from sum systems and those arising from pairs of $C_0$-semigroups. We get a new necessary and sufficient condition for them to be of type III, when the associated sum system is of finite index. Using this criterion, we construct examples of type III $E_0$-semigroups, which can not be distinguished from $E_0$-semigroups of type I by the invariants introduced by Boris Tsirelson. Finally, by considering the local von Neumann algebras, and by associating a type III factor to a given type III $E_0$-semigroup, we show that there exist uncountably many type III $E_0$-semigroups in this family, which are mutually non-cocycle conjugate.