Classification of amenable subfactors of type II

Popa, Sorin

doi:10.1007/bf02392646

Cited by 255 publications

(337 citation statements)

References 51 publications

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“…Instead, Theorem 0.1 is in line with the rigidity results established over the last 25 years in von Neumann algebra theory ( [C1,2], [CJ2], [Po8], [CoHa], [GoNe], ) and orbit equivalence ergodic theory ( [Zi], [GeGo], [Po8], [Ga1,2], , [MoSh], [Hj], [PoSa], [Po9]). …”

Section: Pair; (B) H Is Not Virtually Abelian; (C)supporting

confidence: 77%

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Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

2006

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Abstract. We prove that any isomorphism θ : M 0 ≃ M of group measure space II 1 factors, M 0 = L ∞ (X 0 , µ 0 ) ⋊ σ 0 G 0 , M = L ∞ (X, µ) ⋊ σ G, with G 0 an ICC group containing an infinite normal subgroup with the relative property (T) of KazhdanMargulis (i.e. G 0 w-rigid) and σ a Bernoulli shift action of some group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G 0 ≃ G. Moreover, any isomorphism θ of M 0 onto a "corner" pMp of M, for p ∈ M an idempotent, forces p = 1. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a "group measure space version" of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations. Introduction.We continue in this paper the study of rigidity properties of isomorphisms θ : M 0 ≃ M of crossed product II 1 factors initiated in ([Po4]), concentrating here on the "group measure space" case, when the factors M 0 , M involved come from free ergodic measure preserving (m.p.) actions of groups on probability spaces. Similarly to ([Po4]), we assume the "source" factor M 0 comes from an arbitrary free ergodic measure preserving action σ 0 of a "mildly rigid" group G 0 (i.e. having a "large" subgroup with the relative property (T) of Kazhdan-Margulis), while the "target" factor M comes from an action σ satisfying good "deformation+mixing"

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Section: Pair; (B) H Is Not Virtually Abelian; (C)supporting

confidence: 77%

“…the clustering condition on σ), for this result we only assume σ and σ 0 mixing. The proof takes Sections 5 and 6 and uses "local quantization" techniques, in the spirit of (A.1 in [Po9]). …”

Section: Pair; (B) H Is Not Virtually Abelian; (C)mentioning

confidence: 99%

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

2006

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“…By Lemma 4.1, the principal graph of L(1, 1 4 ) is A ∞ , and d L(1,1/4) = 2. It follows from Proposition 3.4 that α L(1, p 2 ) = d L(1, p 2 ) = 2 p + 1 for all nonnegative integer 2 p. Since α f := α L(1,1/4) has minimal index 4, its principal graph are determined in [Goodman et al 1989;Popa 1994]. We now list some properties of α f : …”

Section: Applicationsmentioning

confidence: 99%

Strong additivity and conformal nets

Xu¹

2005

Pacific J. Math.

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We show that the fixed-point subnet of a strongly additive conformal net under the action of a compact group is strongly additive. Using the idea of the proof we define the notion of strong additivity for a pair of conformal nets and we show that a key fact about induction of pairs, proved earlier under the assumption of finite index, can be generalized to strongly additive pairs of conformal nets. These results are used to classify conformal nets of central charge c = 1 that are not necessarily rational and satisfy a spectrum condition.

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“…Much initial subfactor research classified hyperfinite subfactors of small index ([A 1 : A 0 ] ≤ 4) by studying the standard invariant, i.e., the two towers of higher relative commutants (A i ∩ A j ) i=0,1;j≥0 [Ocn88,GdlHJ89,Izu91,Pop94]. This combinatorial data was axiomatized in three slightly different structures: paragroups [Ocn88], λ-lattices [Pop95], and planar algebras [Jon99].…”

Section: Introductionmentioning

confidence: 99%