Cantor-Schroeder-Bernstein quadruples for Banach spaces

Elói Medina Galego. "Cantor-Schroeder-Bernstein quadruples for Banach spaces." Colloquium Mathematicae 111.1 (2008): 105-115. <http://eudml.org/doc/283698>.

@article{ElóiMedinaGalego2008,
abstract = {Two Banach spaces X and Y are symmetrically complemented in each other if there exists a supplement of Y in X which is isomorphic to some supplement of X in Y. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) Problem for Banach spaces by constructing two non-isomorphic Banach spaces which are symmetrically complemented in each other. In this paper, we show how to modify such a symmetry in order to ensure that X is isomorphic to Y. To do this, first we introduce the notion of Cantor-Schroeder-Bernstein Quadruples for Banach spaces. Then we characterize them by using some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997. This new insight into the geometry of Banach spaces complemented in each other leads naturally to the Strong Square-hyperplane Problem which is closely related to the Schroeder-Bernstein Problem.},
author = {Elói Medina Galego},
journal = {Colloquium Mathematicae},
keywords = {Pełczyński’s decomposition method; Schröder-Bernstein problem},
language = {eng},
number = {1},
pages = {105-115},
title = {Cantor-Schroeder-Bernstein quadruples for Banach spaces},
url = {http://eudml.org/doc/283698},
volume = {111},
year = {2008},
}

TY - JOUR
AU - Elói Medina Galego
TI - Cantor-Schroeder-Bernstein quadruples for Banach spaces
JO - Colloquium Mathematicae
PY - 2008
VL - 111
IS - 1
SP - 105
EP - 115
AB - Two Banach spaces X and Y are symmetrically complemented in each other if there exists a supplement of Y in X which is isomorphic to some supplement of X in Y. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) Problem for Banach spaces by constructing two non-isomorphic Banach spaces which are symmetrically complemented in each other. In this paper, we show how to modify such a symmetry in order to ensure that X is isomorphic to Y. To do this, first we introduce the notion of Cantor-Schroeder-Bernstein Quadruples for Banach spaces. Then we characterize them by using some Banach spaces constructed by W. T. Gowers and B. Maurey in 1997. This new insight into the geometry of Banach spaces complemented in each other leads naturally to the Strong Square-hyperplane Problem which is closely related to the Schroeder-Bernstein Problem.
LA - eng
KW - Pełczyński’s decomposition method; Schröder-Bernstein problem
UR - http://eudml.org/doc/283698
ER -