Witnessing the lack of the Grothendieck property in C ( K )-spaces via convergent sequences
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Witnessing the lack of the Grothendieck property in C(K )-spaces via convergent sequences Jerzy Kakol ¸ 1,2 · Aníbal Moltó3 Received: 20 May 2020 / Accepted: 25 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract 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 1 (K )-Grothendieck space but not a Grothendieck space exactly when the spaces C p (K ) and C p (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 c0 , then C(K ) fails to be 1 (K )-Grothendieck if and only if the topologies of pointwise convergence in C p (K ) and C p (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 C p (K × L) and C p (G) are linearly isomorphic. This extends a remarkable theorem of Cembranos and Freniche. Some examples illustrating the above results are provided. Keywords Grothendieck space · Josefson-Nissenzweig property · (Complemented) copy of c0 Mathematics Subject Classification 46E10 · 54C35
ˇ project 20-22230L and RVO: 67985840. The research for the first named author is supported by the GACR The second author has been supported by a grant of the Ministerio de Ciencia, Innovación y Universidades, PGC2018-094431-B-100. The authors thank to J. C. Ferrando and T. Kania for their suggestions, and also to the referees for their valuable comments and remarks.
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Aníbal Moltó [email protected] Jerzy K¸akol [email protected]
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Faculty of Mathematics and Computer Science, A. Mickiewicz University, Pozna´n, Poland
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Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic
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Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad de Valencia, Dr. Moliner 50, 46100 Burjassot (Valencia), Spain 0123456789().: V,-vol
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J. K¸ akol, A. Moltó
1 Introduction In 1953 Grothendieck proved that every weak* convergent sequence (x n∗ ) in the dual of E := ∞ () converges with respect to the weak topology, i. e. if x n∗ ( f ) → 0 for each f ∈ E, then x ∗∗ (xn∗ ) → 0 for each x ∗∗ ∈ E ∗∗ , where E ∗∗ means the bidual of E. Banach spaces E for which every weak∗ convergent sequence in the dual E ∗ of E weakly converges are called Grothendieck spaces or Banach spaces with the Grothendieck property. A separable Banach space is Grothendieck if and only if it is reflexive, in particular the sequence Banach space c0 is not a Grothendieck space. Closed vector subspaces Y of Grothendieck spaces need not be Grothendieck, although this property is preserved by complemented subspaces Y . Hence, the Banach space C(K ) of continuous real-valued functions over an infinite compact space K equipped with the supremum norm is not a Grothendieck space whenever it contains a complemented copy of c0 . Moreover, Grothendieck observed that every C(K ) over an e
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