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

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

2010-06-18

arxiv.org/abs/1006.3689v2

Adv. Math. 228 (2011), 764-802

Mathematics

Operator Algebras

34 pages

Scientific paper

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.

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