Grothendieck C(K)-spaces and the Josefson-Nissenzweig theorem

0579686 - MÚ 2024 RIV PL eng J - Journal Article
Kąkol, Jerzy - Sobota, D. - Zdomskyy, L.
Grothendieck C(K)-spaces and the Josefson-Nissenzweig theorem.
Fundamenta Mathematicae. Roč. 263, č. 2 (2023), s. 105-131. ISSN 0016-2736. E-ISSN 1730-6329
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : Josefson-Nissenzweig theorem * Grothendieck spaces * Grothendieck property
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Limited access
https://doi.org/10.4064/fm218-6-2023

For a compact space K, the Banach space C(K) is said to have the l(1)-Grothendieck property if every weak* convergent sequence (mu(n) : n is an element of omega) of functionals on C(K) such that mu(n) is an element of l(1)(K) for every n is an element of omega is weakly convergent. Thus, the l(1)- Grothendieck property is a weakening of the standard Grothendieck property for Banach spaces of continuous functions. We observe that C(K) has the l(1)-Grothendieck property if and only if there does not exist any sequence of functionals (mu(n) : n is an element of omega) on C(K), with mu(n) is an element of l(1)(K) for every n is an element of omega, satisfying the conclusion of the classical Josefson-Nissenzweig theorem. We construct an example of a separable compact space K such that C(K) has the l(1)-Grothendieck property but it does not have the Grothendieck property. We also show that for many classical consistent examples of Efimov spaces K their Banach spaces C(K) do not have the l(1)-Grothendieck property.
Permanent Link: https://hdl.handle.net/11104/0348497