On ℓ 1 -preduals distant by 1

Łukasz Piasecki

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2018)

  • Volume: 72, Issue: 2
  • ISSN: 0365-1029

Abstract

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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.

How to cite

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Łukasz Piasecki. "On $\ell _1$-preduals distant by 1." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 72.2 (2018): null. <http://eudml.org/doc/290769>.

@article{ŁukaszPiasecki2018,
abstract = {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.},
author = {Łukasz Piasecki},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Banach-Mazur distance; nearly (almost) isometric Banach spaces; $\ell _1$-preduals, hyperplanes in c, weak$^*$ fixed point property; stable weak$^*$ fixed point property; almost stable weak$^*$ fixed point property; nonexpansive mappings},
language = {eng},
number = {2},
pages = {null},
title = {On $\ell _1$-preduals distant by 1},
url = {http://eudml.org/doc/290769},
volume = {72},
year = {2018},
}

TY - JOUR
AU - Łukasz Piasecki
TI - On $\ell _1$-preduals distant by 1
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2018
VL - 72
IS - 2
SP - null
AB - 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.
LA - eng
KW - Banach-Mazur distance; nearly (almost) isometric Banach spaces; $\ell _1$-preduals, hyperplanes in c, weak$^*$ fixed point property; stable weak$^*$ fixed point property; almost stable weak$^*$ fixed point property; nonexpansive mappings
UR - http://eudml.org/doc/290769
ER -

References

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  9. Goebel, K., Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990. 
  10. Japon-Pineda, M. A., Prus, S., Fixed point property for general topologies in some Banach spaces, Bull. Austral. Math. Soc. 70 (2004), 229-244. 
  11. Michael, E., Pełczyński, A., Separable Banach spaces which admit l n ∞ approximations, Israel J. Math. 4 (1966), 189-198. 
  12. Lazar, A. J., Lindenstrauss, J., On Banach spaces whose duals are L 1 spaces, Israel J. Math. 4 (1966), 205-207. 
  13. 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. 
  14. Piasecki, Ł., On Banach space properties that are not invariant under the Banach-Mazur distance 1, J. Math. Anal. Appl. 467 (2018), 1129-1147. 

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