The radial masa in a free group factor is maximal injective

Cameron, Jan; Fang, Junsheng; Ravichandran, Mohan; White, Stuart

doi:10.1112/jlms/jdq052

Cited by 37 publications

(42 citation statements)

References 22 publications

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“…and since ε is arbitrary, (iv) is an easy consequence of this. So it is enough to prove (5). Let T ∈ (π 1 ⊕ π 2 )(A) ′ , by the Double Commutant Theorem it is enough to show that…”

Section: Definition Of 1-bounded Entropy and Some Technical Lemmasmentioning

confidence: 96%

1-Bounded entropy and regularity problems in von Neumann algebras

Hayes

2016

Int Math Res Notices

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Abstract. We define and investigate the singular subspace Hs(N ⊆ M ) of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N -N subbimodule of L 2 (M ) containing the quasinormalizer introduced in [38], the one-sided quasinormalizer introduced in [11], and the wq-normalizer introduced in [14] (following upon work in [26] and [40]). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N ⊆ M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras in [53] we show that there can be no diffuse, hyperfinite subalgebra of L(Fn) which is spectrally regular. Our techniques are robust enough to repeat this process by transfinite induction and rule out chains of spectrally regular inclusions of algebras starting from a diffuse, hyperfinite subalgebra and ending in L(Fn). We use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II 1 -factors (see [14]). Our results may be regarded as a consistency check for the possibility of existence of a "good" cohomology theory of II 1 -factors. We can also use our techniques to show that if Ut is a one-parameter orthogonal group on a real Hilbert space H and the spectral measure of its generator is singular with respect to the Lebesgue measure, then the continuous core of the free Araki-Woods factor Γ(H, Ut) ′′ is not isomorphic to L(Ft⊗B(ℓ 2 (N)) for any t ∈ (1, ∞]. In particular, Γ(H, Ut) ′′ ∼ = Γ(L 2 (R, m), λt) ′′ where m is Lebesgue measure and λ is the left regular representation. This was previously only know when the spectral measure of the generator of Ut had all of its convolution powers singular with respect to Lebesgue measure. We give similar applications to crossed products by free Bogoliubov actions in the spirit of [22].

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“…and since ε is arbitrary, (iv) is an easy consequence of this. So it is enough to prove (5). Let T ∈ (π 1 ⊕ π 2 )(A) ′ , by the Double Commutant Theorem it is enough to show that…”

Section: Definition Of 1-bounded Entropy and Some Technical Lemmasmentioning

confidence: 96%

1-Bounded entropy and regularity problems in von Neumann algebras

Hayes

2016

Int Math Res Notices

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“…In [CFRW10], it is proved that the subalgebra of the free group factor generated by the symmetric laplacian operator (the radial subalgebra) is maximal amenable. In [Ho14], C. Houdayer provided uncountably many non-isomorphic examples of abelian maximal amenable subalgebras in II 1 -factors.…”

Section: Introductionmentioning

confidence: 99%

Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups

Boutonnet

Carderi

2015

Geom. Funct. Anal.

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Abstract. We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms of Λ-invariant measures on some compact Γ-space. The strategy of proof is different from S. Popa's approach to maximal amenability via central sequences [Po83], and relies on elementary computations in a crossed-product C * -algebra.

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“…He defined the notion of asymptotic orthogonality property (AOP) and showed that a singular maximal abelian subalgebra (masa) with the AOP is maximal amenable. Many other examples have been given using the same strategy [Ge96,She06,CFRW10,Bro14,Hou14a,BCa]. A completely new strategy in proving maximal amenability has been given in [BCb].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%