Ultraproducts, QWEP von Neumann algebras,
and the Effros–Maréchal topology

Ando, Hiroshi; Haagerup, Uffe; Winsløw, Carl

doi:10.1515/crelle-2014-0005

Cited by 39 publications

(21 citation statements)

References 17 publications

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“…Indeed, to enter the micro-society (or institution) of mathematicians, one needs to make contact with works from current mathematical research, and subsequently get to question it to the point of contributing to such works. A very long sequence of works needs to be studied to get from didactic texts such as (Eilers et al, 2015) to current works like (Ando, Haagerup, & Winsløw, 2016). This, at least, is the kind of reason which the institution would give for the size of O to be studied in five years, and in particular for the size of O to be studied in the first half of the second semester.…”

Section: To Whom Is Questioning Important and Why?mentioning

confidence: 99%

Questioning definitions at university: the case of analysis

Winsløw

2019

EMP

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Section: To Whom Is Questioning Important and Why?mentioning

confidence: 99%

Questioning definitions at university: the case of analysis

Winsløw

2019

EMP

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“…Lemma 4.1 (cf. [Co1,Theorem 2.7], [AHW,Lemma 4.8]). Let (M, H, J, P ) be a standard form of a σ-finite von Neumann algebra M. Let ξ 0 ∈ P be a cyclic and separating vector.…”

Section: σ-Finite Von Neumann Algebrasmentioning

confidence: 99%

Haagerup Approximation Property for Arbitrary von Neumann Algebras

Okayasu¹,

Tomatsu²

2015

Publ. Res. Inst. Math. Sci.

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We attempt presenting a notion of the Haagerup approximation property for an arbitrary von Neumann algebra by using its standard form. We also prove the expected heredity results for this property.Ruan and Q. Xu in [JRX] in terms of non-commutative L p -spaces. In particular, the non-commutative L 2 -spaces become standard forms, and hence their result is a generalization of her characterization of semidiscrete von Neumann algebras. Therefore it immediately follows that the injectivity implies the Haagerup approximation property in our sense.The Haagerup approximation property has various stabilities. Among them, we will prove the following result (Theorem 5.9): Theorem B. Let N ⊂ M be an inclusion of von Neumann algebras. Suppose that there exists a norm one projection from M onto N. If M has the Haagerup approximation property, then so does N. In [CS], M. Caspers and A. Skalski independently introduce the notion of the Haagerup approximation property. Our formulation actually coincides with theirs because in either case, the Haagerup approximation property is preserved under taking the crossed products by R-actions. (See Remark 5.8.)This paper is organized as follows: In Section 2, the basic notions are reviewed and we introduce the Haagerup approximation property for a von Neumann algebra. In Section 3, we study some permanence properties such as reduced von Neumann algebras, tensor products, the commutant and the direct sums. In Section 4, we consider the case where M is a σ-finite von Neumann algebra with a faithful normal state ϕ. We present the proof of Theorem A. We also discuss the free product of von Neumann algebras and examples. In Section 5, we study the crossed product of a von Neumann algebra by a locally compact group. We show that a von Neumann algebra has the Haagerup approximation property if and only if so does its core von Neumann algebra. The proof of Theorem B is presented.Acknowledgements. The authors are grateful to Narutaka Ozawa for various useful comments on our work. Theorem B is the answer to his question to us. The first author would like to thank Marie Choda and Yoshikazu Katayama for fruitful discussions. The authors also express their gratitude to the referees for several helpful comments and revisions. DefinitionWe first fix notations and recall basic facts. Let M be a von Neumann algebra. We denote by M sa and M + , the set of all self-adjoint elements and all positive elements in M, respectively. We also denote by M * and M + * the space of all normal linear functionals and all positive normal linear functionals on M, respectively.Let us recall the definition of a standard form of a von Neumann algebra that is formulated by Haagerup in [Ha1].Definition 2.1. Let (M, H, J, P ) be a quadruple, where M is a von Neumann algebra, H is a Hilbert space on which M acts, J is a conjugate-linear isometry on H with J 2 = 1 H , and P ⊂ H is a closed convex cone which is self-dual, i.e., P = {ξ ∈ H | ξ, η ≥ 0 for η ∈ P }.3

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“…KEP asks if every C *algebra embeds into an ultrapower of the Cuntz algebra O 2 . 1 The Kirchberg Embedding Problem was studied model theoretically by the first author and Sinclair in [11]. In that paper, KEP is shown to be equivalent to the statement that there is a C * -algebra that is both nuclear and existentially closed.…”

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“…Further model-theoretic equivalents of KEP were given by the first author in [9], where Goldbring was partially supported by NSF grant DMS-2054477. 1 Incidentally, one might ask if there is an "infinite", that is, type III, von Neumann algebra analog of CEP. For example, one might ask if every von Neumann algebra embeds with expectation into the Ocneanu ultrapower of the hyperfinite III 1 factor R ∞ .…”

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confidence: 99%

“…For example, one might ask if every von Neumann algebra embeds with expectation into the Ocneanu ultrapower of the hyperfinite III 1 factor R ∞ . In [1], Ando, Haagerup, and Winslow prove that this question is equivalent to CEP itself (and thus has a negative answer). it was shown that KEP is equivalent to the statement that O 2 is the enforceable C * -algebra.…”

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