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**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).

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).

Reviewer: V.Panchapagesan Thiruyaiyaru (Merida)

### MSC:

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

### Keywords:

limited set; \(\omega^*\) zero-sequence; \(\omega^*\)-sequentially precompact subset; \(GP\)-space; Gelfand-Phillips-property### Citations:

Zbl 0365.46020
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 |

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