Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences

0531560 - MÚ 2021 RIV ES eng J - Journal Article
Kąkol, Jerzy - Moltó, A.
Witnessing the lack of the Grothendieck property in C(K)-spaces via convergent sequences.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Roč. 114, č. 4 (2020), č. článku 179. ISSN 1578-7303. E-ISSN 1579-1505
R&D Projects: GA ČR(CZ) GF20-22230L
Institutional support: RVO:67985840
Keywords : (Complemented) copy of c * Grothendieck space * Josefson-Nissenzweig property
OECD category: Pure mathematics
Impact factor: 2.169, year: 2020
Method of publishing: Limited access
https://doi.org/10.1007/s13398-020-00914-3

Let K be a compact Hausdorff space and let C(K) be the space of all scalar-valued, continuous functions on K. We show that C(K) is an ℓ1(K) -Grothendieck space but not a Grothendieck space exactly when the spaces Cp(K) and Cp(K⊕ N#) are not linearly isomorphic, where N# is the one-point compactificiation of the discrete space of natural numbers. (That is, if C(K) contains a complemented copy of c, then C(K) fails to be ℓ1(K) -Grothendieck if and only if the topologies of pointwise convergence in Cp(K) and Cp(K⊕ N#) are linearly isomorphic.) Moreover, for infinite compact spaces K and L, there exists a compact space G that has a non-trivial convergent sequence and such that Cp(K× L) and Cp(G) are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided.
Permanent Link: http://hdl.handle.net/11104/0310190