Fullness, Connes’ $\chi $-groups, and ultra-products of amalgamated free products over Cartan subalgebras

Ueda, Yoshimichi

doi:10.1090/s0002-9947-02-03100-8

Cited by 19 publications

(23 citation statements)

References 28 publications

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“…The aim of this paper is to investigate factoriality, Connes' type III invariants and fullness of arbitrary AFP von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on AFP von Neumann algebras [46][47][48][49][50] and we obtain new examples of full AFP factors for which we can explicitely compute Connes' type III invariants. Let us point out that the questions of factoriality, Connes' type III invariants and fullness for plain free product von Neumann algebras were completely settled by Ueda in [48,49].…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 68%

“…In order to understand the position of centralizing nets in M = M 1 * B M 2 , we use a completely different method based on Popa's deformation/rigidity theory. As we will see, our method allows us to generalize many of the results obtained by Ueda in [46][47][48][49][50]. We proceed in two steps.…”

Section: Fullness Of Afp Von Neumann Algebrasmentioning

confidence: 96%

“…In order to understand the position of centralizing nets in [46][47][48][49][50] generalized Popa's asymptotic orthogonality property [39] for free group factors to AFP von Neumann algebras. For this method to work though, one needs to assume not only that M 1 M1 B but also that M 1 has 'many' unitaries that are perpendicular to B with respect to E 1 : M 1 → B and that these unitaries lie in the centralizer of a fixed faithful normal state on M 1 .…”

Section: Fullness Of Afp Von Neumann Algebrasmentioning

confidence: 99%

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Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

Houdayer¹,

Isono²

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We investigate factoriality, Connes' type III invariants and fullness of arbitrary amalgamated free product von Neumann algebras using Popa's deformation/rigidity theory. Among other things, we generalize many previous structural results on amalgamated free product von Neumann algebras and we obtain new examples of full amalgamated free product factors for which we can explicitely compute Connes' type III invariants.Note that the inclusion of continuous cores c(P ) ⊂ c(M ) is the one associated with a fixed faithful normal conditional expectation E P : M → P . The proof of Theorem B generalizes the one of [Io12, Theorem 6.3] and relies on a combination of Popa's deformation/rigidity theory and Connes-Tomita-Takesaki modular theory.Secondly, to understand the position of centralizing nets in M = M 1 * B M 2 , we apply Theorem B to P = M . We want to use Assertion (i) and exploit the key fact that M ∨ B ω (resp. c(M ) ∨ c(B) ω ) can be regarded as an AFP von Neumann algebra. Assuming that M 1 M 1 B and the inclusion B ⊂ M 2 is entirely nontrivial as in Theorem A, we prove that Assertion (iii) in Theorem B cannot occur. On the other hand, we improve Assertion (ii) in Theorem B by showing that c(M 1 ) c(M 1 ) c(B). However, as we already pointed out, this does not contradict our assumption that M 1 M 1 B. Thus, in order to obtain the fullness property for AFP von Neumann algebras, we need to add extra assumptions.When the amalgam B is a type I von Neumann algebra, we obtain the following sharp rigidity result for the asymptotic centralizer M ω , generalizing [Ue12, Theorems 4.8 and 4.10].

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Section: Introduction and Statement Of The Main Resultssupporting

confidence: 68%

Section: Fullness Of Afp Von Neumann Algebrasmentioning

confidence: 96%

Section: Fullness Of Afp Von Neumann Algebrasmentioning

confidence: 99%

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Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

Houdayer¹,

Isono²

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…It is natural as a next project to consider the same questions for more general amalgamated free product von Neumann algebras. Such attempts were already made by us [21,24,23] almost 10 years ago for amalgamated free products over Cartan subalgebras. However the results there are far from satisfactory as compared to those on plain free product von Neumann algebras.…”

Section: Introductionmentioning

confidence: 99%

Some analysis of amalgamated free products of von Neumann algebras in the non-tracial setting

Ueda

2013

Journal of the London Mathematical Society

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Several techniques together with some partial answers are given to the questions of factoriality, type classification and fullness for amalgamated free product von Neumann algebras.This fact is easily confirmed in the exactly same way as in [27, Lemma 2.1]. It is clear that E • E M k = E holds. Consider the natural inclusion of the so-called continuous cores:which is independent of the choice of χ thanks to Connes's Radon-Nikodym cocycle theorem. The canonical liftings (still being faithful normal conditional expectations) E :4) Remark that the original E and E k are recovered as the restrictions of E and E k to M and M k via the canonical embeddings M ֒→ M and M k ֒→ M k , respectively. Here is a simple but important fact [21, Theorem 5.1] that M 1 and M 2 are freely independent with amalgamation over N with respect to E, and moreover M = M 1 ∨ M 2 . Consequently the following natural formula holds: ( M , E) = ( M 1 , E 1 ) ⋆ N ( M 2 , E 2 ). (2.5) The canonical faithful normal semifinite traces Tr M , Tr M k (k = 1, 2) and Tr N on M , M k and N , respectively, (see [20, Theorem XII.1.1]) must satisfy Tr M = Tr N • E and Tr M k = Tr N • E k (see e.g. [14, §4]).

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“…Let α : Z D be the Bernoulli shift and let θ : G D be the diagonal action of θ 0 . Then θ canonically extends to M := D ⋊ α * α F 2 , which is a non-McDuff factor of type II 1 (see, for example, Theorem 10 of Ueda [28]). Then this action has the Rohlin property.…”

mentioning

confidence: 99%

Locally Compact Separable Abelian Group Actions on Factors with the Rokhlin Property

Shimada¹

2014

Publ. Res. Inst. Math. Sci.

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Fullness, Connes’ $\chi $-groups, and ultra-products of amalgamated free products over Cartan subalgebras

Cited by 19 publications

References 28 publications

Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras

Some analysis of amalgamated free products of von Neumann algebras in the non-tracial setting

Locally Compact Separable Abelian Group Actions on Factors with the Rokhlin Property

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