Dual spaces of compact operator spaces and the weak Radon-Nikodým property

Keun Young Lee

Displaying similar documents to “Dual spaces of compact operator spaces and the weak Radon-Nikodým property”

Existence theorem for the Hammerstein integral equation

Mieczysław Cichoń, Ireneusz Kubiaczyk (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we prove an existence theorem for the Hammerstein integral equation $x (t) = p (t) + λ \int_{I} K (t, s) f (s, x (s)) d s$ , where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

Remarks on continuous images of Radon-Nikodým compacta

Marián J. Fabián, Martin Heisler, Eva Matoušková (1998)

Commentationes Mathematicae Universitatis Carolinae

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A family of compact spaces containing continuous images of Radon-Nikod’ym compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continuous image of a Radon-Nikod’ym compact $K$ we prove: If $K$ is totally disconnected, then it is Radon-Nikod’ym compact. If $K$ is adequate, then it is even Eberlein compact.

Weak Baire measurability of the balls in a Banach space

José Rodríguez (2008)

Studia Mathematica

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Let X be a Banach space. The property (∗) “the unit ball of X belongs to Baire(X, weak)” holds whenever the unit ball of X* is weak*-separable; on the other hand, it is also known that the validity of (∗) ensures that X* is weak*-separable. In this paper we use suitable renormings of $ℓ^{\infty} (ℕ)$ and the Johnson-Lindenstrauss spaces to show that (∗) lies strictly between the weak*-separability of X* and that of its unit ball. As an application, we provide a negative answer to a question raised...

The relation between the dual and the adjoint Radon transforms

Cnops, J.

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[For the entire collection see Zbl 0742.00067.]Let $P^{m}$ be the set of hyperplanes $σ : 〈 \vec{x}, \vec{θ} 〉 = p$ in $ℛ^{m}$ , $S^{m - 1}$ the unit sphere of $ℛ^{m}$ , $E^{m}$ the exterior of the unit ball, $T^{m}$ the set of hyperplanes not passing through the unit ball, $R f (\vec{θ}, p) = \int_{σ} f (\vec{x}) d \vec{x}$ the Radon transform, $R^{#} g (\vec{x}) = \int_{S^{m - 1}} g (\vec{θ}, 〈 \vec{x}, \vec{θ} 〉) d S_{\vec{θ}}$ its dual. $R$ as operator from $L^{2} (ℛ^{m})$ to $L^{2} (S^{m - 1)} \times ℛ)$ is a closable, densely defined operator, $R^{*}$ denotes the operator given by $(R^{*} g) (\vec{x}) = R^{#} g (\vec{x})$ if the integral exists for $\vec{x} \in ℛ^{m}$ a.e. Then the closure of $R^{*}$ is the adjoint of $R$ . The author shows that the Radon transform and its dual can be linked by...

Metric spaces admitting only trivial weak contractions

Richárd Balka (2013)

Fundamenta Mathematicae

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If (X,d) is a metric space then a map f: X → X is defined to be a weak contraction if d(f(x),f(y)) < d(x,y) for all x,y ∈ X, x ≠ y. We determine the simplest non-closed sets X ⊆ ℝⁿ in the sense of descriptive set-theoretic complexity such that every weak contraction f: X → X is constant. In order to do so, we prove that there exists a non-closed $F_{σ}$ set F ⊆ ℝ such that every weak contraction f: F → F is constant. Similarly, there exists a non-closed $G_{δ}$ set G ⊆ ℝ such that every weak...

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Adam Osękowski (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $ℙ (s u p_{t \geq 0} | Y_{t} | \geq 1) \leq 3.375 . . . ∥ X ∥ ₁$ . As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

Proper cocycles and weak forms of amenability

Paul Jolissaint (2015)

Colloquium Mathematicae

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Let G and H be locally compact, second countable groups. Assume that G acts in a measure class preserving way on a standard space (X,μ) such that $L^{\infty} (X, μ)$ has an invariant mean and that there is a Borel cocycle α: G × X → H which is proper in the sense of Jolissaint (2000) and Knudby (2014). We show that if H has one of the three properties: Haagerup property (a-T-menability), weak amenability or weak Haagerup property, then so does G. In particular, we show that if Γ and Δ are measure equivalent...

Approximate and weak amenability of certain Banach algebras

P. Bharucha, R. J. Loy (2010)

Studia Mathematica

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The notions of approximate amenability and weak amenability in Banach algebras are formally stronger than that of approximate weak amenability. We demonstrate an example confirming that approximate weak amenability is indeed actually weaker than either approximate or weak amenability themselves. As a consequence, we examine the (failure of) approximate amenability for $ℓ^{p}$ -sums of finite-dimensional normed algebras.

Every Radon-Nikodym Corson compact space is Eberlein compact

J. Orihuela, W. Schachermayer, M. Valdivia (1991)

Studia Mathematica

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On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

Weak Distances between Random Subproportional Quotients of $ℓ ₁^{m}$

Piotr Mankiewicz (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

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Lower estimates for weak distances between finite-dimensional Banach spaces of the same dimension are investigated. It is proved that the weak distance between a random pair of n-dimensional quotients of $ℓ ₁^{n ²}$ is greater than or equal to c√(n/log³n).

Weak Type Inequality for the Square Function of a Nonnegative Submartingale

Adam Osękowski (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be a nonnegative submartingale and S(f) denote its square function. We show that for any λ > 0, $λ ℙ (S (f) \geq λ) \leq π / 2 ∥ f ∥ ₁$ , and the constant π/2 is the best possible. The inequality is strict provided ∥f∥₁ ≠ 0.

On weak supercyclicity II

Carlos S. Kubrusly, Bhagwati P. Duggal (2018)

Czechoslovak Mathematical Journal

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This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly $l$ -sequentially supercyclic, and (iii) weak $l$ -sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space...

Weak dimensions and Gorenstein weak dimensions of group rings

Yueming Xiang (2021)

Czechoslovak Mathematical Journal

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Let $K$ be a field, and let $G$ be a group. In the present paper, we investigate when the group ring $K [G]$ has finite weak dimension and finite Gorenstein weak dimension. We give some analogous versions of Serre’s theorem for the weak dimension and the Gorenstein weak dimension.

The weak Phillips property

Ali Ülger (2001)

Colloquium Mathematicae

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Let X be a Banach space. If the natural projection p:X*** → X* is sequentially weak*-weak continuous then the space X is said to have the weak Phillips property. We present several characterizations of the spaces having this property and study its relationships to other Banach space properties, especially the Grothendieck property.

Convex Corson compacta and Radon measures

Grzegorz Plebanek (2002)

Fundamenta Mathematicae

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Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ (ℝ^{ω ₁})$ , and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$ -points.

Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska (2015)

Studia Mathematica

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Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for...