Strongly solid group factors which are not interpolated free group factors

Houdayer, Cyril

doi:10.1007/s00208-009-0417-6

Cited by 16 publications

(14 citation statements)

References 37 publications

(87 reference statements)

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“…For instance, it was proven in [20,Theorem B] that when the orthogonal representation π : Z → O(H R ) is mixing, the crossed product II 1 factor Γ(H R ) ′′ ⋊ π Z is strongly solid. This gave new examples of strongly solid II 1 factors which are not * -isomorphic to interpolated free group factors (see also [17]).…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Houdayer

2014

Advances in Mathematics

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In this paper, we investigate several structural properties for crossed product II1 factors M arising from free Bogoljubov actions associated with orthogonal representations π : G → O(HR) of arbitrary countable discrete groups. Under fairly general assumptions on the orthogonal representation π : G → O(HR), we show that M does not have property Gamma of Murray and von Neumann. Then we show that any regular amenable subalgebra A ⊂ M can be embedded into L(G) inside M . Finally, when G is assumed to be amenable, we locate precisely any possible amenable or Gamma extension of L(G) inside M .2010 Mathematics Subject Classification. 46L10; 46L54; 46L55; 22D25.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Houdayer

2014

Advances in Mathematics

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“…In this paper, we use a combination of Popa's deformation and intertwining techniques [16,18,19] and the techniques of Ozawa and Popa [14,15] to give another example of a strongly solid II 1 factor not isomorphic to an amplification of a free group factor, i.e. to an interpolated free group factor [5,21] (the first example of this kind was constructed by the first-named author in [12], answering an open question of Popa [17]).…”

Section: Introductionmentioning

confidence: 99%

“…The proof of Theorems A and B, following a "deformation/rigidity" strategy, is a combination of the ideas and techniques in [12,14,15,18]. We will use the "free malleable deformation" by automorphisms (α t , β) defined on Γ(H R ) ′′ * Γ(H R ) ′′ = Γ(H R ⊕ H R ) ′′ .…”

Section: Introductionmentioning

confidence: 99%

Strongly Solid II1 Factors with an Exotic MASA

Houdayer¹,

Shlyakhtenko²

2010

International Mathematics Research Notices

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Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid II1 factor M containing an "exotic" maximal abelian subalgebra A: as an A,A-bimodule, L 2 (M ) is neither coarse nor discrete. Thus we show that there exist II1 factors with such property but without Cartan subalgebras. It also follows from Voiculescu's free entropy results that M is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.2000 Mathematics Subject Classification. 46L10; 46L54.

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“…We remark that as a corollary to the techniques used in the proof of Theorem 0.2, we are able to strengthen a result of Houdayer [9] on free product group factors admitting no Cartan subalgebras.…”

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confidence: 58%

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

mentioning

confidence: 98%

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 group factors which are not interpolated free group factors

Cited by 16 publications

References 37 publications

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Strongly Solid II1 Factors with an Exotic MASA

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 group factors which are not interpolated free group factors

Cited by 16 publications

References 37 publications

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Structure of II 1 factors arising from free Bogoljubov actions of arbitrary groups

Strongly Solid II1 Factors with an Exotic MASA

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
0
0
0
View full text Add to dashboard Cite