Finite von Neumann Algebras and Masas

Sinclair, Allan M.; Smith, Roger

doi:10.1017/cbo9780511666230

Cited by 117 publications

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“…The von Neumann algebra M, e B generated by M and e B is called the basic construction, which plays a crucial role in the study of von Neumann subalgebras of finite von Neumann algebras. The basic construction has many remarkable properties (see [9,12,20]). In particular, there exists a unique faithful tracial weight Tr on M, e B such that…”

Section: Preliminariesmentioning

confidence: 99%

“…(see [20,Lemma 4.3.4,Theorem 4.3.11]). An examination of the proof of [20,Lemma 4.3.4] shows that we may construct the index set I to have a minimal element i = 1 and we may take ξ 1 to be ξ.…”

Section: Preliminariesmentioning

confidence: 99%

“…This property was introduced by Robertson, Sinclair and the third author [19,18,20] for masas (maximal abelian subalgebras) of type II 1 factors. They showed that if B has the weak asymptotic homomorphism property, then B is singular in M , and the purpose of introducing this property was to have an easily verifiable criterion for singularity.…”

Section: Introductionmentioning

confidence: 99%

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The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite Von Neumann Algebras

2011

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A triple of finite von Neumann algebras B ⊆ N ⊆ M is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operatorsfor all x, y ∈ M . We prove that a triple of finite von Neumann algebras B ⊆ N ⊆ M has the relative weak asymptotic homomorphism property if and only if N contains the set of all x ∈ M such that Bx ⊆ n i=1 x i B for a finite number of elements x 1 , . . . , x n in M . Such an x is called a one sided quasi-normalizer of B, and the von Neumann algebra generated by all one sided quasi-normalizers of B is called the one sided quasi-normalizer algebra of B. We characterize one sided quasi-normalizer algebras for inclusions of group von Neumann algebras and use this to show that one sided quasi-normalizer algebras and quasi-normalizer algebras are not equal in general. We also give some applications to inclusions L(H) ⊆ L(G) arising from containments of groups. For example, when L(H) is a masa we determine the unitary normalizer algebra as the von Neumann algebra generated by the normalizers of H in G.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Relative Weak Asymptotic Homomorphism Property for Inclusions of Finite Von Neumann Algebras

2011

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“…Recall that the von Neumann algebra VN(G) is hyperfinite if and only if G is amenable [SS,Theorem 3.8.2]. A Fourier multiplier is a normal linear map T : VN(G) → VN(G) such that there exists a function ϕ : G → C such that for any g ∈ G we have T (λ g ) = ϕ g λ g .…”

Section: Preliminariesmentioning

confidence: 99%

Analytic semigroups on vector valued noncommutative L^p-spaces

Arhancet

2013

Studia Math.

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We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or a R-analytic semigroup (Tt ⊗ IdE) t 0 on the vector valued noncommutative L p -space L p (M, E). Moreover, we give applications to the H ∞ (Σ θ ) functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu. IntroductionIt is shown in [JMX] that whenever (T t ) t 0 is a noncommutative diffusion semigroup on a von Neumann algebra M equipped with a faithful normal state such that each T t satisfies the Rota dilation property, then the negative generator of itsis the open sector of angle 2θ around the positive real axis (0, +∞). Our first principal result is an extension of this theorem to the vector valued case. We use a different approach using R-analyticity instead of square functions.In order to describe our result, we need several definitions. (1) T is completely positivewhere (σ φ t ) t∈R and (σ ψ t ) t∈R denote the automorphism groups of the states φ and ψ respectively.In particular, when (M, φ) = (N, ψ), we say that T is a φ-Markov map.A linear map T : M → N satisfying conditions (1) − (3) above is normal. If, moreover, condition (4) is satisfied, then it is known that there exists a unique completely positive, unital mapThe next definition is a variation of the one of [AnD] (see also [Ric] and [HaM] Neumann algebra P with QWEP equipped with a faithful normal state χ , and * -monomorphisms J 0 : N → P and J 1 : M → P such that J 0 is (φ, χ)-Markov and J 1 is (ψ, χ)-Markov, satisfying, moreover, T = J * 0 • J 1 . We say that T is hyper-factorizable if the same property is true with a hyperfinite von Neumann algebra P .

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“…The first two axioms, then require this state to be a faithful trace. The effect of the third axiom, is to ensure that the image of the unit ball of M under the GNS representation corresponding to this faithful trace is closed in the weak operator topology (see for example [24,Lemma A.3.3] …”

Section: Introductionmentioning

confidence: 99%