On a class of $\mathrm{II}_1$ factors with at most one Cartan subalgebra

By Nathanial P. Brown Narutaka Ozawa, Sorin Popa. 2007

216 citations recorded by the source corpus.

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L(\Bbb F_r)$ generates an amenable von Neumann subalgebra. Moreover, any II$_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group $\Bbb F_r$, $2\leq r \leq \infty$, on a probability space $(X,\mu)$ is profinite then the group measure space factor $L^\infty(X)\rtimes \Bbb F_r$ has unique Cartan subalgebra, up to unitary conjugacy.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/593afd861d64df22735bfcac8db3d8527a24f647

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Adrian Ioana
2. S. Vaes
3. Sorin Popa
4. Thomas Sinclair
5. I. Chifan
6. A. Furman
7. K. De Commer
8. Bogdan Udrea

In Mathematical Physics

1. Adrian Ioana
2. S. Vaes
3. Sorin Popa
4. Thomas Sinclair
5. I. Chifan
6. A. Furman
7. Bogdan Udrea
8. K. De Commer

Citations in the corpus

• Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, unknown): “iodic. (3) M is a type III1 factor if and only if (Ut) is nonperiodic and nontrivial. Let us start by recalling some fundamental structural results for free group factors. In their breakthrough paper [27], Ozawa and Popa showed that the free group factors L(Fn) are strongly solid, i.e. the normalize”
• Classification and rigidity for von Neumann algebras (2012, methodology): “ile II1 factors coming from icc groups Γ admitting a proper cocycle into ℓ2(Γ) were shown to be solid in [Pe06]. For further examples of prime and solid II1 factors, see [Oz04, Po06a, CI08, CH08]. In [OP07], Ozawa and Popa discovered that the free group factors enjoy a remarkable structural property”
• W*-superrigidity for Bernoulli actions of property (T) groups (2010, methodology): “Since Γ2 is non–amenable, this contradicts [28], Corollary 2.”
• A random matrix approach to the Peterson-Thom conjecture (2020, background): “It would mean that not only can we find a pair of homomorphisms satisfying the conclusion of Jung’s theorem, but that a randomly chosen pair satisfies Jung’s theorem.”
• ON THE STRUCTURAL THEORY OF II1 FACTORS OF NEGATIVELY CURVED GROUPS, II. ACTIONS BY PRODUCT GROUPS (2011, methodology): “En devéloppant une nouvelle approche, qui combine les méthodes de Peterson [29], d’Ozawa et Popa [27, 28], et d’Ozawa [25], nous renforçons ce résultat en montrant que ce facteur est fortement solide.”
• Examples of factors which have no Cartan subalgebras (2012, unknown): “d relied on the free probability theory. Shlyakhtenko also proved that free Araki–Woods factors of type IIIλ have no Cartan subalgebras [26]. In the rigidity theory, Ozawa and Popa gave first examples [20]. They proved that the free group factors are such examples, and they actually proved that these”
• CCAP for Universal Discrete Quantum Groups (2013, methodology): “From the viewpoint of deformation rigidity theory, W ∗CCAP and its variants are exploited to show the sparsity of Cartan subalgebras in the breakthrough papers by Ozawa and Popa [29], and by Popa and Vaes [31].”
• Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups (2012, background): “ny direct product Γ = H×Gbetween a non-amenable group Gand a certain type of locally finite, infinite, non-commutative group H, is not C-rigid. Another class of groups that are not C-rigid was found in [OP08] and it consists of certain semidirect products Γ = H⋊G, with Habelian, notably Γ = Z2 ⋊SL(2,Z”
• Easy quantum groups and quantum subgroups of a semi-direct product quantum group (2013, background): “An important notion in von Neumann algebra theory is strong solidity introduced in [OP10].”
• Cartan subalgebras of amalgamated free product II$_1$ factors (2012, background): “ot have a Cartan subalgebra. So far, our results only apply to Cartan subalgebras of amalgamated free product von Neumann algebras M = M1 ∗B M2. From now on, we more generally study, in the spirit of [OP07] and [PV11], normalizers of arbitrary diffuse amenable von Neumann subalgebras A⊂ M. Recall tha”