On peak phenomena for non-commutative H ∞

Ueda, Yoshimichi

doi:10.1007/s00208-008-0277-5

Cited by 17 publications

(46 citation statements)

References 23 publications

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“…has received much attention in Banach space theory, and indeed many serious investigations were carried out; see, for example, [17; 20, § 6.d]. The present notes are part of our attempts, started in [26], to give more 'functional analysis insight' to many theorems obtained in those investigations on L 1 (T)/H ∞ (D) ⊥ by discussing them in some non-commutative set-up.…”

Section: Introductionmentioning

confidence: 93%

“…Via the embedding H ∞ (D) has the 'standard' predualhas received much attention in Banach space theory, and indeed many serious investigations were carried out; see, for example, [17; 20, § 6.d]. The present notes are part of our attempts, started in [26], to give more 'functional analysis insight' to many theorems obtained in those investigations on L 1 (T)/H ∞ (D) ⊥ by discussing them in some non-commutative set-up.Natural non-commutative generalizations of H ∞ (D) were introduced by Arveson [4] in the 1960s under the name of maximal subdiagonal algebras, and here we call them noncommutative H ∞ -algebras. The finite tracial ones have been well studied so that we mainly deal with the finite tracial non-commutative H ∞ -algebras in the present notes.…”

mentioning

confidence: 96%

“…The second purpose is to give some supplements to our previous notes [26]. Firstly, we briefly explain how the results in [26] can easily be generalized to the case of semifinite tracial noncommutative H ∞ -algebras A = H ∞ (M, Tr) with semifinite M and faithful normal semifinite tracial weights Tr. Secondly, we discuss the results in [3] that still remain to be proved in the non-commutative set-up.…”

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confidence: 99%

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On the predual of non-commutative H ^∞

Ueda

2011

Bulletin of the London Mathematical Society

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The liftability property of weakly relatively compact subsets in M /A ⊥ to M is established for any non-commutative H ∞ -algebra A = H ∞ (M, τ ). Some supplementary results to our previous works are also given. IntroductionLet H ∞ (D) be the Banach algebra of all bounded analytic functions on the open unit disc D equipped with the supremum norm − ∞ , and it is known that H ∞ (D) is faithfully embedded into L ∞ (T) by taking non-tangential limits. Via the embedding H ∞ (D) has the 'standard' predualhas received much attention in Banach space theory, and indeed many serious investigations were carried out; see, for example, [17; 20, § 6.d]. The present notes are part of our attempts, started in [26], to give more 'functional analysis insight' to many theorems obtained in those investigations on L 1 (T)/H ∞ (D) ⊥ by discussing them in some non-commutative set-up.Natural non-commutative generalizations of H ∞ (D) were introduced by Arveson [4] in the 1960s under the name of maximal subdiagonal algebras, and here we call them noncommutative H ∞ -algebras. The finite tracial ones have been well studied so that we mainly deal with the finite tracial non-commutative H ∞ -algebras in the present notes. Let M be a finite von Neumann algebra with a faithful normal tracial state τ . A σ-weakly closed unital (not necessarily self-adjoint) subalgebra A of M is called a finite tracial non-commutative H ∞ -algebra, which we denote by A = H ∞ (M, τ ), if A + A * is σ-weakly dense in M and the unique τ -preserving conditional expectation E : M → D := A ∩ A * is multiplicative on A. Here we define A * = {a * ∈ M | a ∈ A}, while the symbol X has been used as the dual Banach space of a given Banach space X. The reader can find an excellent survey for non-commutative H ∞ -algebras in [21]. It is plain to see that A has the 'standard' predual M /A ⊥ , which is the main object in our study.It is the main purpose of the present notes to prove that any weakly relatively compact subset in M /A ⊥ can be 'lifted up' to a weakly relatively compact subset in M . In particular, it immediately follows that the Mackey topology on A is indeed the relative topology induced from that on M . Hence, this part of the present notes provides a non-self-adjoint generalization of Sakai's result [24] (also Akemann's result [1, Theorem II.7]). Here, recall that this liftability property was already established by many hands, for example, Kisljakov [14], Delbaen [11] and Pe lczyński [17, § 7], in the 1970s for the classical and commutative L 1 (T)/H ∞ (D) ⊥ and its function algebra generalizations (see [17, p. 54] for further information), and it played a key rôle in any existing proof of the fact that L 1 (T)/H ∞ (D) ⊥ has the Dunford-Pettis

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Section: Introductionmentioning

confidence: 93%

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On the predual of non-commutative H ^∞

Ueda

2011

Bulletin of the London Mathematical Society

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“…In a recent paper [7] Blecher and Labuschagne investigated whether every von Neumann algebra verifies Ueda's peak set theorem [21]. It was discovered that the answer turns on the existence of singular states with a certain continuity property.…”

Section: Measurable Cardinalsmentioning

confidence: 99%

Quantum measurable cardinals

Blecher

Weaver

2017

Journal of Functional Analysis

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Abstract. We investigate states on von Neumann algebras which are not normal but enjoy various forms of infinite additivity, and show that these exist on B(H) if and only if the cardinality of an orthonormal basis of H satisfies various large cardinal conditions. For instance, there is a singular countably additive pure state on B(l 2 (κ)) if and only if κ is Ulam measurable, and there is a singular < κ-additive pure state on B(l 2 (κ)) if and only if κ is measurable. The proofs make use of Farah and Weaver's theory of quantum filters [10]. We can generalize some of these characterizations to arbitrary von Neumann algebras. Applications to Ueda's peak set theorem for von Neumann algebras are discussed in the final section.

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“…Then by [Ued,Theorem 2], A has a unique predual, namely A * = M * / A ⊥ . Also, each M n (A) is a subdiagonal operator algebra, so applying [Ued,Corollary 2] …”

Section: One-sided L-embedded Spacesmentioning

confidence: 99%

Operator spaces which are one-sided M-ideals in their bidual

Sharma

2010

Studia Math.

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Abstract. We generalize an important class of Banach spaces, namely the Membedded Banach spaces, to the non-commutative setting of operator spaces. The one-sided M -embedded operator spaces are the operator spaces which are one-sided M -ideals in their second dual. We show that several properties from the classical setting, like the stability under taking subspaces and quotients, unique extension property, Radon Nikodým Property and many more, are retained in the non-commutative setting. We also discuss the dual setting of one-sided L-embedded operator spaces.

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On peak phenomena for non-commutative H ∞

Cited by 17 publications

References 23 publications

On the predual of non-commutative H ^∞

On the predual of non-commutative H ^∞

Quantum measurable cardinals

Operator spaces which are one-sided M-ideals in their bidual

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On peak phenomena for non-commutative H ∞

Cited by 17 publications

References 23 publications

On the predual of non-commutative H ∞

On the predual of non-commutative H ∞

Quantum measurable cardinals

Operator spaces which are one-sided M-ideals in their bidual

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Product

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On the predual of non-commutative H ^∞

On the predual of non-commutative H ^∞