On $\ell _1$-preduals distant by 1

Łukasz Piasecki

ℓ_{1}

-preduals distant by 1

Łukasz Piasecki

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 72 (2018), / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

For every predual $X$ of $ℓ_{1}$ such that the standard basis in $ℓ_{1}$ 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 $ℓ_{1}$ , 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, $σ (Y^{*}, Y)$ -FPP) whenever $d (X, Y) = 1$ . Then, we construct a predual $X$ of $ℓ_{1}$ for which $ℓ_{1}$ lacks the stable $σ (ℓ_{1}, X)$ -FPP but it has almost stable $σ (ℓ_{1}, X)$ -FPP, which in turn is a strictly stronger property than the $σ (ℓ_{1}, X)$ -FPP. Finally, in the general setting of preduals of $ℓ_{1}$ , we give a sufficient condition for almost stable weak $^{*}$ fixed point property in $ℓ_{1}$ and we prove that for a wide class of spaces this condition is also necessary.

Publié le : 2018-01-01
EUDML-ID : urn:eudml:doc:290769

@article{bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_41,
     author = {\L ukasz Piasecki},
     title = {On $\ell \_1$-preduals distant by 1},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {72},
     year = {2018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_41}
}

Łukasz Piasecki. On $\ell _1$-preduals distant by 1. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 72 (2018) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2018_72_2_41/

Bibliographie

Alspach, D. E., A $ℓ_{1}$ -predual which is not isometric to a quotient of $C (α)$ , 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 $ℓ_{1}$ , 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 $ℓ_{1}$ , J. Math. Anal. Appl. 452 (1) (2017), 673-684.

Casini, E., Miglierina, E., Piasecki, Ł., Popescu, R., Weak $^{*}$ fixed point property in $ℓ_{1}$ 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.