Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors

TL;DR

This paper proves that all free Araki-Woods factors have the complete metric approximation property and lack Cartan subalgebras in nonamenable subfactors, distinguishing them from certain type III_1 factors.

Contribution

It establishes the approximation property for free Araki-Woods factors and shows they do not contain Cartan subalgebras, answering a longstanding question about their classification.

Findings

All free Araki-Woods factors have the complete metric approximation property.

Nonamenable subfactors with normal conditional expectations lack Cartan subalgebras.

Type III_1 factors by Connes are not isomorphic to free Araki-Woods factors.

Abstract

We show that all the free Araki-Woods factors $\Gamma(H_\R, U_t)"$ have the complete metric approximation property. Using Ozawa-Popa's techniques, we then prove that every nonamenable subfactor $\mathcal{N} \subset \Gamma(H_\R, U_t)"$ which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type ${\rm III_1}$ factors constructed by Connes in the '70s can never be isomorphic to any free Araki-Woods factor, which answers a question of Shlyakhtenko and Vaes.