Abstract
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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We thank the referee for a careful reading of the manuscript and for providing an interesting solution to Problem 8.
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Communicated by: Yasuyuki Kawahigashi
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Research of M. González was partially supported by MICINN (Spain), Grant PID2019-103961GB-C22. T. Kania acknowledges with thanks funding received from SONATA 15 No. 2019/35/D/ST1/01734.
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González, M., Kania, T. Grothendieck spaces: the landscape and perspectives. Jpn. J. Math. 16, 247–313 (2021). https://doi.org/10.1007/s11537-021-2116-3
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DOI: https://doi.org/10.1007/s11537-021-2116-3
Keywords and phrases
- Grothendieck space
- G-space
- property (V)
- property (V ∞)
- Banach space
- Grothendieck operator
- C*-algebra
- Banach lattice
- positive Grothendieck property
- quantitative Grothendieck property
- pseudo-intersection number
- locally convex Grothendieck space
- twisted sum
- push-out