# Arveson's characterisation of CCR flows: the multiparameter case

## TL;DR

This paper extends Arveson's characterization of CCR flows to the multiparameter setting, providing new insights into their structure, examples, and classification criteria.

## Contribution

It establishes that multiparameter CCR flows are characterized by decomposability and the existence of a unit, and explores their properties and classifications in detail.

## Key findings

- Uncountably many decomposable E0-semigroups lack units.
- CCR flows associated with pure isometric representations determine the unitary class of the representation.
- Conditions for CCR flows to be prime are identified.

## Abstract

In this paper, we revisit Arveson's characterisation of CCR flows in terms of decomposibility of the product system in the multiparameter context. We show that a multiparameter $E_0$-semigroup is a CCR flow if and only if it is decomposable and admits a unit. In contrast to the one parameter situtation, we exhibit uncountably many examples of decomposable $E_0$-semigroups which do not admit any unit. As applications, we show that for a pure isometric representation $V$, the associated CCR flow $\alpha^{V}$ remembers the unitary equivalence class of $V$. We also compute the positive contractive local cocycles and projective local cocycles of a CCR flow. A necessary and a sufficient condition for a CCR flow to be prime is obtained.