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Grothendieck spaces: the landscape and perspectives
- Source :
- Japanese Journal of Mathematics, 2021. 16(2), 247-313, UCrea Repositorio Abierto de la Universidad de Cantabria, Universidad de Cantabria (UC)
- Publication Year :
- 2021
- Publisher :
- arXiv, 2021.
-
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 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.<br />Comment: 55 pp., accepted for publication in Japanese Journal of Mathematics
- Subjects :
- Positive Grothendieck property
Pure mathematics
Property (philosophy)
Primary 46A35, 46B20, Secondary 46L10
General Mathematics
Quantitative Grothendieck property
Banach space
Mathematical proof
C*-algebra
Push-out
Grothendieck operator
Development (topology)
Pseudo-intersection number
FOS: Mathematics
Operator Algebras (math.OA)
Mathematics - General Topology
Grothendieck space
Property (V∞)
Mathematics
Property (V)
Dual space
Locally convex Grothendieck space
Mathematics - Operator Algebras
General Topology (math.GN)
G-space
State (functional analysis)
Banach lattice
Connection (mathematics)
Functional Analysis (math.FA)
Mathematics - Functional Analysis
Twisted sum
Section (category theory)
Subjects
Details
- Database :
- OpenAIRE
- Journal :
- Japanese Journal of Mathematics, 2021. 16(2), 247-313, UCrea Repositorio Abierto de la Universidad de Cantabria, Universidad de Cantabria (UC)
- Accession number :
- edsair.doi.dedup.....f33ad41fe13966c87b73605d57a088c5
- Full Text :
- https://doi.org/10.48550/arxiv.2102.03838