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2018 | 72 | 2 |
Tytuł artykułu

On \(\ell_1\)-preduals distant by 1

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.
Rocznik
Tom
72
Numer
2
Opis fizyczny
Daty
wydano
2018
online
2018-12-22
Twórcy
Bibliografia
  • Alspach, D. E., A \(\ell_1\)-predual which is not isometric to a quotient of \(C(\alpha)\), arXiv:math/9204215v1 [math.FA] 27 Apr. 1992.
  • Banach, S., Theorie des operations lineaires, Monografie Matematyczne, Warszawa, 1932.
  • Cambern, M., On mappings of sequence spaces, Studia Math. 30 (1968), 73-77.
  • Casini, E., Miglierina, E., Piasecki, Ł., Hyperplanes in the space of convergent sequences and preduals of \(\ell_1\), Canad. Math. Bull. 58 (2015), 459-470.
  • Casini, E., Miglierina, E., Piasecki, Ł., Separable Lindenstrauss spaces whose duals lack the weak\(^*\) fixed point property for nonexpansive mappings, Studia Math. 238 (1) (2017), 1-16.
  • Casini, E., Miglierina, E., Piasecki, Ł., Popescu, R., Stability constants of the weak\(^*\) fixed point property in the space \(\ell_1\), J. Math. Anal. Appl. 452 (1) (2017), 673-684.
  • Casini, E., Miglierina, E., Piasecki, Ł., Popescu, R., Weak\(^*\) fixed point property in \(\ell_1\) and polyhedrality in Lindenstrauss spaces, Studia Math. 241 (2) (2018), 159-172.
  • Casini, E., Miglierina, E., Piasecki, Ł., Vesely, L., Rethinking polyhedrality for Lindenstrauss spaces, Israel J. Math. 216 (2016), 355-369.
  • Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990.
  • Japon-Pineda, M. A., Prus, S., Fixed point property for general topologies in some Banach spaces, Bull. Austral. Math. Soc. 70 (2004), 229-244.
  • Michael, E., Pełczyński, A., Separable Banach spaces which admit \(l_n^\infty\) approximations, Israel J. Math. 4 (1966), 189-198.
  • Lazar, A. J., Lindenstrauss, J., On Banach spaces whose duals are \(L_1\) spaces, Israel J. Math. 4 (1966), 205-207.
  • Pełczyński, A., in collaboration with Bessaga, Cz., Some aspects of the present theory of Banach spaces, in: Stefan Banach Oeuvres. Vol. II, PWN, Warszawa, 1979, 221-302.
  • Piasecki, Ł., On Banach space properties that are not invariant under the Banach-Mazur distance 1, J. Math. Anal. Appl. 467 (2018), 1129-1147.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_41
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