Cantor–Schroeder–Bernstein quadruples for Banach spaces

Elói Medina Galego¹

¹ Department of Mathematics – IME University of São Paulo São Paulo 05315-970, Brazil

Colloquium Mathematicum, Tome 111 (2008) no. 1, pp. 105-115.

Résumé

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.

Détail

DOI : 10.4064/cm111-1-10

Keywords: banach spaces symmetrically complemented each other there exists supplement which isomorphic supplement gowers solved schroeder bernstein cantor bernstein problem banach spaces constructing non isomorphic banach spaces which symmetrically complemented each other paper modify symmetry order ensure isomorphic first introduce notion cantor schroeder bernstein quadruples banach spaces characterize using banach spaces constructed gowers maurey insight geometry banach spaces complemented each other leads naturally strong square hyperplane problem which closely related schroeder bernstein problem