A $C(K)$ Banach space which does not have the Schroeder–Bernstein property

Piotr Koszmider¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8, P.O. Box 21 00-956 Warszawa, Poland and Instytut Matematyki Politechniki Łódzkiej Wólczańska 215 90-924 Łódź, Poland and Departamento de Análisis Matemático Facultad de Ciencias, Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 212 (2012) no. 2, pp. 95-117

Résumé

We construct a totally disconnected compact Hausdorff space $K_+$ which has clopen subsets $K_+^{\prime\prime}\subseteq K_+^{\prime}\subseteq K_+$ such that $K_+^{\prime\prime}$ is homeomorphic to $K_+$ and hence $C(K_+^{\prime\prime})$ is isometric as a Banach space to $C(K_+)$ but $C(K_+^{\prime})$ is not isomorphic to $C(K_+)$. This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form $C(K)$ which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder–Bernstein problem for Banach spaces of the form $C(K)$. The subset $K_+$ is obtained as a particular compactification of the pairwise disjoint union of an appropriately chosen sequence $(K_{1,n}\cup K_{2,n})_{n\in \mathbb N}$ of $K$s for which $C(K)$s have few operators. We have $K_+^{\prime}=K_+\setminus K_{1,0}$ and $K_+^{\prime\prime}=K_+\setminus (K_{1,0}\cup K_{2,0}).$

Détail

DOI : 10.4064/sm212-2-1

Keywords: construct totally disconnected compact hausdorff space which has clopen subsets prime prime subseteq prime subseteq prime prime homeomorphic hence prime prime isometric banach space prime isomorphic gives nonisomorphic banach spaces necessarily nonseparable form which isomorphic complemented subspaces each other even above strong isometric sense providing solution schroeder bernstein problem banach spaces form subset obtained particular compactification pairwise disjoint union appropriately chosen sequence cup mathbb which have few operators have prime setminus prime prime setminus cup