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On a theorem of Sobczyk

Published online by Cambridge University Press:  17 April 2009

Aníbal Moltó
Aníbal Moltó
Affiliation:
Departamento de Análisis Matematico, Universitat de Valencia, Dr. Moliner 50 46100 Burjassot, Valencia, Spain
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Abstract

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In this paper the result of Sobczyk about complemented copies of c0 is extended to a class of Banach spaces X such that the unit ball of their dual endowed with the weak* topology has a certain topological property satisfied by every Corson-compact space. By means of a simple example it is shown that if Corson-compact is replaced by Rosenthal-compact, this extension does not hold. This example gives an easy proof of a result of Phillips and an easy solution to a question of Sobczyk about the existence of a Banach space E, c0 ⊂ E ⊂ l∞, such that E is not complemented in l∞ and c0 is not complemented in E. Assuming the continuum hypothesis, it is proved that there exists a Rosenthal-compact space K such that C(K) has no projectional resolution of the identity.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 43 , Issue 1 , February 1991 , pp. 123 - 130
Copyright
Copyright © Australian Mathematical Society 1991

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