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

By Cyril Houdayer, 'Eric Ricard. 2010

58 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a17189636c1e25f507a36bc0874e6aa205d3e5ec

Citations in the corpus

• Examples of factors which have no Cartan subalgebras (2012, unknown): “is, φ= φ◦ E. To prove Theorem A, we need only slight modifications of the proof of (a special case of) [27,Theorem 3.1]. Theorem B can be proved by seeing its continuous core, and this idea comes from [16] and [17]. Since condition (AOC)+ is similar to condition (AO) with respect to the continuous co”
• Approximation properties for free orthogonal and free unitary quantum groups (2011, unknown): “1), and it is also a useful tool for proving other approximation properties, such as the weak∗-(completely) bounded approximation property (see [29, 30]).”