Quantitative Grothendieck property

Bendová, Hana

doi:10.1016/j.jmaa.2013.11.033

Cited by 11 publications

(15 citation statements)

References 19 publications

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“…We say that X is c-Grothendieck if δ w (x * n ) ≤ cδ w * (x * n ) whenever (x * n ) n∈N is a bounded sequence in X * . It is known that ℓ ∞ is even 1-Grothendieck due to H. Bendová [1,Theorem 1.1]. We generalize this result on a wider class of spaces.…”

Section: Introduction and Main Resultsmentioning

confidence: 58%

1-Grothendieck C(K) spaces

Lechner

2017

Journal of Mathematical Analysis and Applications

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A Banach space is said to be Grothendieck if weak and weak * convergent sequences in the dual space coincide. This notion has been quantificated by H. Bendová. She has proved that ℓ ∞ has the quantitative Grothendieck property, namely, it is 1-Grothendieck. Our aim is to show that Banach spaces from a certain wider class are 1-Grothendieck, precisely, C(K) is 1-Grothendieck provided K is a totally disconnected compact space such that its algebra of clopen subsets has the so called Subsequential completeness property.2010 Mathematics Subject Classification. 46E15; 54G05; 46A20.

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Section: Introduction and Main Resultsmentioning

confidence: 58%

1-Grothendieck C(K) spaces

Lechner

2017

Journal of Mathematical Analysis and Applications

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“…This notion was introduced by V.P. Fonf and J. Lindenstrauss in [8] and studied by several authors (see, e.g., [2,14,15,18] and the references therein). Let us recall the following so-called Boundary Theorem (for a relatively simple proof, see [18,Theorem 2]).…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…Theorem 7.6). Proposition 7.4 characterizes property (U) in terms of intermediate envelope as well as of the so-called 1-Grothendieck property (a quantitative version of the Grothendieck property, introduced in [2]). Combining our characterization and the results in [14], we obtain that ∞ has property (U).…”

Section: Introductionmentioning

confidence: 99%

Intersection properties of the unit ball

Bernardi

Veselý

2019

Journal of Mathematical Analysis and Applications

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Let X be a real Banach space with the closed unit ball B X and the dual X * . We say that X has the intersection property (I) (general intersection property (GI), respectively) if, for each countable family (for each family, respectively) {B i } i∈A of equivalent closed unit balls such thatis the bipolar set of B i , that is, the bidual unit ball corresponding to B i . In this paper we study relations between properties (I) and (GI), and geometric and differentiability properties of X. For example, it follows by our results that if X is Fréchet smooth or X is a polyhedral Banach space then X satisfies property (GI), and hence also property (I). Moreover, for separable spaces X, properties (I) and (GI) are equivalent and they imply that X has the ball generated property. However, properties (I) and (GI) are not equivalent in general. One of our main results concerns C(K) spaces: under certain topological condition on K, satisfied for example by all zerodimensional compact spaces and hence by all scattered compact spaces, we prove that C(K) satisfies (I) if and only if every nonempty G δ -subset of K has nonempty interior.2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B04, 46B10, 46B25.

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“…One possible quantification of the Grothendieck property has already been studied in [3] and [9]. Let us remind the definition:…”

Section: Relation To the Grothendieck Propertymentioning

confidence: 99%

C*-algebras have a quantitative version of Pełczyński's property (V)

Krulišová

2017

Czech.Math.J.

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A Banach space X has Pe lczyński's property (V) if for every Banach space Y every unconditionally converging operator T : X → Y is weakly compact. H. Pfitzner proved that C * -algebras have Pe lczyński's property (V). In the preprint [8] the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner's theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.

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Quantitative Grothendieck property

Cited by 11 publications

References 19 publications

1-Grothendieck C(K) spaces

1-Grothendieck C(K) spaces

Intersection properties of the unit ball

C*-algebras have a quantitative version of Pełczyński's property (V)

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