Strongly Solid II1 Factors with an Exotic MASA

Houdayer, Cyril; Shlyakhtenko, Dimitri

doi:10.1093/imrn/rnq117

Cited by 25 publications

(49 citation statements)

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“…We can use our results to give other examples of algebras M with h(M ) ≤ 0. In many cases by results of [22] as well as [21] these algebras have the complete metric approximation property and are strongly solid. Thus they share many properties with free group factors, but are not isomorphic to them.…”

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confidence: 98%

“…Thus µ is a measure in the same absolute continuity class as the spectral measure of U C . Since µ ∈ ℓ p (Z), we know that π is mixing and hence by [22] we know that Γ q (H) ⋊ απ Z is strongly solid and has CMAP. As µ is singular with respect to Lebesgue measure we have h(Γ q (H) ⋊ απ Z) = 0 and thus Γ q (H) ⋊ απ Z is not isomorphic to an interpolated free group factor.…”

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confidence: 99%

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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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confidence: 98%

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confidence: 99%

1-Bounded entropy and regularity problems in von Neumann algebras

Hayes

2016

Int Math Res Notices

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“…Then the symmetric L ∞ (K, µ)-bimodule H ν given by (1.1) and (1.2) is isomorphic with the symmetric L(G)-bimodule associated, as in Remark 3.5, with the cyclic orthogonal representation of G with spectral measure ν. In particular, as in Remark 3.5, the von Neumann algebras M = Γ(H ν , J ν , L ∞ (K), µ) ′′ can also be realized as a free Bogoljubov crossed product by the countable abelian group G. In this way, Proposition 7.3 generalizes the results of [HS09,Ho12a]. Note however that for a free Bogoljubov crossed product M = Γ(K R ) ′′ ⋊ G with G abelian, the subalgebra L(G) ⊂ M is never an s-MASA.…”

Section: Absence Of Cartan Subalgebrasmentioning

confidence: 55%

“…Therefore, our Theorem A is a generalization of similar results proved in [Ho12b] for free Bogoljubov crossed products. Although free Bogoljubov crossed products M = L(F ∞ )⋊G with G abelian provide examples of MASAs L(G) ⊂ M with interesting properties (see [HS09,Ho12a]), L(G) ⊂ M can never be an s-MASA (see Remark 7.4).…”

Section: Introductionmentioning

confidence: 99%

Thin II1 factors with no Cartan subalgebras

Krogager¹,

Vaes²

2019

Kyoto J. Math.

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It is a wide open problem to give an intrinsic criterion for a II 1 factor M to admit a Cartan subalgebra A. When A ⊂ M is a Cartan subalgebra, the A-bimodule L 2 (M ) is "simple" in the sense that the left and right action of A generate a maximal abelian subalgebra of B(L 2 (M )). A II 1 factor M that admits such a subalgebra A is said to be s-thin. Very recently, Popa discovered an intrinsic local criterion for a II 1 factor M to be s-thin and left open the question whether all s-thin II 1 factors admit a Cartan subalgebra. We answer this question negatively by constructing s-thin II 1 factors without Cartan subalgebras.

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“…Other examples of strongly solid factors were subsequently constructed by Houdayer [9] and by Houdayer and Shlyakhtenko [10].…”

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Strong solidity of group factors from lattices in SO(n,1) and SU(n
Sinclair
¹

2011
Journal of Functional Analysis
29
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We show that the group factors LΓ , where Γ is an ICC lattice in either SO(n, 1) or SU(n, 1), n 2, are strongly solid in the sense of Ozawa and Popa (2010) [13]. This strengthens a result of Ozawa and Popa (2010) [14] showing that these factors do not have Cartan subalgebras.In their breakthrough paper [13], Ozawa and Popa brought new techniques to bear on the study of free group factors which allowed them to show that these factors possess a powerful structural property, what they called "strong solidity." Definition 0.1. (See Ozawa and Popa [13].) A II 1 factor M is strongly solid if for any diffuse amenable subalgebra P ⊂ M we have that N M (P ) is amenable.As usual, N M (P ) = {u ∈ U(M): uP u * = P } denotes the normalizer of P in M. It can be seen that every nonamenable II 1 subfactor of a strongly solid II 1 factor is non-Gamma, prime and has no Cartan subalgebras. Thus, Ozawa and Popa's result broadened and offered a unified approach to the two main results on the structure of free group factors hitherto known: Voiculescu's [29]

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Strongly Solid II1 Factors with an Exotic MASA

Cited by 25 publications

References 29 publications

1-Bounded entropy and regularity problems in von Neumann algebras

1-Bounded entropy and regularity problems in von Neumann algebras

Thin II1 factors with no Cartan subalgebras

Strong solidity of group factors from lattices in SO(n,1) and SU(n
Sinclair
¹

2011
Journal of Functional Analysis
29
0
0
0
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Strongly Solid II1 Factors with an Exotic MASA

Cited by 25 publications

References 29 publications

1-Bounded entropy and regularity problems in von Neumann algebras

1-Bounded entropy and regularity problems in von Neumann algebras

Thin II1 factors with no Cartan subalgebras

Strong solidity of group factors from lattices in SO(n,1) and SU(nSinclair1 2011Journal of Functional Analysis29000View full textAdd to dashboardCite

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Strong solidity of group factors from lattices in SO(n,1) and SU(n
Sinclair
¹

2011
Journal of Functional Analysis
29
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0
0
View full text Add to dashboard Cite