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

Tom 212 / 2012

Piotr Koszmider Studia Mathematica 212 (2012), 95-117 MSC: 46B03, 46E15, 06E15, 54G20. DOI: 10.4064/sm212-2-1

Streszczenie

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 $K_+^{\prime\prime}=K_+\setminus (K_{1,0}\cup K_{2,0}).$

Autorzy

  • Piotr KoszmiderInstitute 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
    e-mail

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