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

Abstract

We show that all the free Araki-Woods factors $\Gamma (H_{\mathbb{R}},U_t)"$ have the complete metric approximation property. Using Ozawa-Popa's techniques, we then prove that every nonamenable subfactor $\mathcal{N}\subset \Gamma (H_{\mathbb{R}},U_t)"$ which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type $\mathrm{II}{\mathrm{I}}_1$ 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.

34 pages

References in corpus (3)

• Non injectivity of the q-deformed von Neumann algebra
• Strongly solid $\mathrm{I}{\mathrm{I}}_1$ factors with an exotic MASA
• On multiplicity and free absorption for free Araki-Woods factors

Cited by in corpus (26)

• CCAP for universal discrete quantum groups
• Factoriality, type classification and fullness for free product von Neumann algebras
• A class of $\mathrm{I}{\mathrm{I}}_1$ factors with an exotic abelian maximal amenable subalgebra
• Amalgamated free product type III factors with at most one Cartan subalgebra
• Structure of $\mathrm{I}{\mathrm{I}}_1$ factors arising from free Bogoljubov actions of arbitrary groups
• Asymptotic structure of free product von Neumann algebras
• Asymptotic structure of free Araki-Woods factors
• Strong solidity of free Araki-Woods factors
• On the factoriality of q-deformed Araki-Woods von Neumann algebras
• Classification of a family of non almost periodic free Araki-Woods factors
• Reconstruction of twisted Steinberg algebras
• Explicit examples of equivalence relations and factors with prescribed fundamental group and outer automorphism group
• Complete metric approximation property for $q$-Araki-Woods algebras
• Radial multipliers on reduced free products of operator algebras
• Structure of extensions of free Araki-Woods factors
• Examples of factors which have no Cartan subalgebras
• Haagerup approximation property for arbitrary von Neumann algebras
• Pointwise convergence of noncommutative Fourier series
• Some prime factorization results for free quantum group factors
• Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors
• Thin II_1 factors with no Cartan subalgebras
• Unique prime factorization for infinite tensor product factors
• A note on weak amenability for reduced free products of discrete quantum groups
• On bi-exactness of discrete quantum groups
• Weak Exactness for C*-algebras and Application to Condition (AO)
• Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras