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A survey on Banach spaces C(K) with few operators

Una revisión de los espacios de Banach C(K) con pocos operadores

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

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=gI+S or T*=g*I+S where g is continuous on K, g* is Borel on K and S are 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. Borodulin-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.

Resumen

Se dice que un espacio de Banach C(K), formado por las funciones continuas definidas en un espacio compacto de Haussdorf K y con valores reales, provisto con la norma supremo, tiene pocos operadores si para cada operador T en C(K) se tiene que T=gI+S o bien T*=g*I+S, donde la función g: K→K es continua, g*: K→K es una función de Borel y S es débilmente compacto en C(K) o C*(K), respectivamente. Los espacios C(K) con pocos operadores comparten algunas de sus propiedades con los espacios de Gowers y Maurey, pero su norma es la más sencilla posible. Por ejemplo, algunos de ellos son espacios de Banach que no se pueden descomponer y no son isomorfos a sus hiperplanos. Los espacios de Banach de funciones continuas con pocos operadores han proporcionado soluciones a varios problemas de espacios de Banach que han estado abiertos mucho tiempo. Esta clase de espacios está siendo paulatinamente más comprendida y aplicada en artículos recientes de P. Borodulin-Nadzieja, R. Fajardo, V. Ferenczi, E. Medina Galego, M. Martín, J. Merí, G. Plebanek, I. Schlackow y del autor. Aquí describimos propiedades básicas, aplicaciones y relevantes problemas abiertos relativos a esta clase de espacios de Banach.

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  1. Instytut Matematyki Politechniki Łódzkiej, ul. Wólczańska 215, 90-924, Łódź, Poland

    Piotr Koszmider

  2. Departamento de Análisis Matemático Facultad de Ciencias, Universidad de Granada, 18071, Granada, Spain

    Piotr Koszmider

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Koszmider, P. A survey on Banach spaces C(K) with few operators. RACSAM 104, 309–326 (2010). https://doi.org/10.5052/RACSAM.2010.19

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