Free Araki-Woods factors and Connes' bicentralizer problem

TL;DR

This paper proves that free Araki-Woods factors of type III_1 have trivial bicentralizer, and demonstrates the existence of a faithful state with trivial centralizer in the algebra, advancing understanding of Connes' bicentralizer problem.

Contribution

It establishes the triviality of the bicentralizer for all free Araki-Woods factors of type III_1, providing new insights into their structure and Connes' problem.

Findings

Trivial bicentralizer for free Araki-Woods factors of type III_1

Existence of a faithful state with trivial centralizer

Advancement in understanding Connes' bicentralizer problem

Abstract

We show that for any free Araki-Woods factor $\mathcal{M} = \Gamma(H_\R, U_t)"$ of type ${\rm III_1}$, the bicentralizer of the free quasi-free state $\varphi_U$ is trivial. Using Haagerup's Theorem, it follows that there always exists a faithful normal state $\psi$ on $\mathcal{M}$ such that $(\mathcal{M}^\psi)' \cap \mathcal{M} = \C$.