A survey on Banach spaces \(C(K)\) with few operators. (English) Zbl 1227.46021
Summary: We say that a Banach space \(C(K)\) of all real-valued continuous functions on a compact Hausdorff space \(K\) with the supremum norm has few operators if, for every linear bounded operator \(T\) on \(C(K)\), we have \(T=gl+S\), or \(T^*=g^*l+S\), where \(g\) is continuous on \(K\), \(g^*\) is Borel on \(K\), and \(S\) is weakly compact on \(C(K)\) or \(C^*(K)\), respectively. \(C(K)\) spaces with few operators share some properties with the spaces of Gowers and Maurey, but their norm is as simple as possible. For example, some of them are indecomposable Banach spaces and are not isomorphic to their hyperplanes. Banach spaces of continuous functions with few operators provided solutions to several long standing open problems in the theory of Banach spaces. This class of spaces is being gradually illuminated and applied further in the recent work of P. Borodutin-Nadzieja, R. Fajardo, V. Ferenczi, E. Medina Galego, M. Martín, J. Merí, G. Plebanek, I. Schlackow and the author. We describe basic properties, applications and relevant open problems concerning this class of Banach spaces.
MSC:
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 46B26 | Nonseparable Banach spaces |
| 54G99 | Peculiar topological spaces |
| 03E35 | Consistency and independence results |
Keywords:
spaces with few operators; indecomposable Banach spaces; Banach spaces of continuous functionsReferences:
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