On C(K) Grothendieck spaces

Barcenas, Diomedes; Mármol, Luis Gerardo

doi:10.1007/bf02874635

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…In general, convergence in the strong operator topology does not imply convergence in the operator norm. However for Grothendieck C(K)-spaces the following result is available ( [35]; see also [11,Theorem 2.1]). Proposition 3.1.3.…”

Section: Characterisations and First Propertiesmentioning

confidence: 99%

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Grothendieck spaces: the landscape and perspectives

Kania¹

2021

Preprint

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In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between Grothendieck spaces (Banach spaces for which weak-and weak*-sequential convergences in the dual space coincide) and vector measures. This connection was developed in his book with J. Uhl Jr. Vector measures. Additionally, Diestel's paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day.The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describing the main examples of spaces with this property, recording the solutions to Diestel's problems, providing generalisations/extensions or new proofs of various results concerning Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory.

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Section: Characterisations and First Propertiesmentioning

confidence: 99%

“…More precisely, can G-spaces be characterised topologically? (11) Let A be a Boolean algebra whose Stone space is a G-space. Does there exist a Boolean subalgebra B ⊂ A whose Stone space K fails to be a G-space yet every weakly* convergent sequence of purely atomic measures on K converges weakly?…”

Section: Ultrapowers and Ultraproductsmentioning

confidence: 99%

Grothendieck spaces: the landscape and perspectives

Kania¹

2021

Preprint

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Grothendieck spaces: the landscape and perspectives

Kania

2021

Jpn. J. Math.

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In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between Grothendieck spaces (Banach spaces for which weak-and weak*-sequential convergences in the dual space coincide) and vector measures. This connection was developed further in his book with J. Uhl Jr. Vector measures. Additionally, Diestel's paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day.The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describing the main examples of spaces with this property, recording the solutions to Diestel's problems, providing generalisations/extensions or new proofs of various results concerning Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory.

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Epic math battle of history: Grothendieck vs Nikodym

Głodkowski,

Widz

2025

Journal of Functional Analysis

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On C(K) Grothendieck spaces

Cited by 3 publications

References 16 publications

Grothendieck spaces: the landscape and perspectives

Grothendieck spaces: the landscape and perspectives

Grothendieck spaces: the landscape and perspectives

Epic math battle of history: Grothendieck vs Nikodym

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