×

Limited sets in \(C(K)\)-spaces and examples concerning the Gelfand- Phillips-property. (English) Zbl 0797.46013

A subset \(A\) of a Banach space \(X\) is said to be limited if each \(\omega^*\) zero-sequence in \(X^*\) converges uniformly on \(A\). The Banach space \(X\) is said to have the Gelfand-Phillips-property \((GP\)- property) if each limited set in \(X\) is relatively weakly compact and in that case, \(X\) is called a \(GP\)-space. The author shows that a \(C(K)\)- space is a \(GP\)-space if all normed sequences \((f_ n)\) in \(C(K)\) of nonnegative functions having pairwise disjoint supports are not limited and also gives a sufficient condition for the above sequences to be limited.
Sharpening a construction in R. Haydon [Isr. J. Math. 28, No. 4, 313-324 (1977; Zbl 0365.46020)], the author constructs a compact \(K\) for which \(C(K)\) is not \(GP\) but contains a subspace isometric to \(c_ 0\) so that \(C(K)/c_ 0\) is \(GP\). It then follows that the \(GP\)-property is not a three space property. The above example is used to give a Banach space \(X \not\supset \ell_ 1\) such that \(X\) is not \(GP\), thereby proving that the \(GP\)-property does not imply that the dual unit ball contains a \(\omega^*\)-sequentially precompact subset which norms \(X\) up to a constant (though the existence of such a subset in the dual unit ball implies that \(X\) is a \(GP\)-space).

MSC:

46B22 Radon-Nikodým, Kreĭn-Milman and related properties

Citations:

Zbl 0365.46020
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bourgain, Limited operators and strict cosingularity, Math, Nachrichten 119 pp 55– (1989) · Zbl 0601.47019
[2] Davis, Factoring weakly compact operators, J. Funct. Anal. 17 pp 311– (1974) · Zbl 0306.46020
[3] Diestel, Sequences and series in Banach spaces 92, in: Graduate Texts in Mathematics (1986)
[4] Drewnowski, On Banach spaces with the Gelfand-Phillips property, Math. Zeitschrift 193 pp 405– (1986) · Zbl 0629.46020
[5] L. Drewnowski G. Emmanuele On Banach spaces with the Gelfand-Phillips property II 1986 · Zbl 0629.46020
[6] Emmanuele, Gelfand-Phillips property in a Banach space of vector valued measures, Math. Nachrichten 127 pp 21– (1986) · Zbl 0622.46026
[7] Hagler, A Banach space not containing l1 whose dual ball is not w* sequentially compact, Israel J. Math. 28 (4) pp 325– (1977)
[8] Hagler, Smoothness and weak* sequential compactness, Proc. Amer. Math. Soc. 78 (4) pp 497– (1980) · Zbl 0463.46010
[9] P. R. Halmos Lectures on Boolean algebras 1 1963
[10] Haydon, On Banach spaces which contain l1({\(\tau\)}) and types of measures on compact spaces, Israel J. Math. 28 (4) pp 313– (1977) · Zbl 0365.46020
[11] Haydon, A non reflexive Grothendieck space that does not contain l, Israel J. Math. 40 (1) pp 65– (1981) · Zbl 1358.46007
[12] T. Schlumprecht Limited sets in Banach spaces 1988
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.