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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Extreme contractions on continuous vector-valued function spaces


Authors: Hasan Al-Halees and Richard J. Fleming
Journal: Proc. Amer. Math. Soc. 134 (2006), 2661-2666
MSC (2000): Primary 47B38, 46E40
Published electronically: March 23, 2006
MathSciNet review: 2213745
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Abstract: An old question asks whether extreme contractions on $ C(K)$ are necessarily nice; that is, whether the conjugate of such an operator maps extreme points of the dual ball to extreme points. Partial results have been obtained. Determining which operators are extreme seems to be a difficult task, even in the scalar case. Here we consider the case of extreme contractions on $ C(K,E)$, where $ E$ itself is a Banach space. We show that every extreme contraction $ T$ on $ C(K,E)$ to itself which maps extreme points to elements of norm one is nice, where $ K$ is compact and $ E$ is the sequence space $ c_{0}$.


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Additional Information

Hasan Al-Halees
Affiliation: Department of Mathematics, Saginaw Valley State University, University Center, Michigan 48710-0001

Richard J. Fleming
Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08282-7
PII: S 0002-9939(06)08282-7
Received by editor(s): March 15, 2005
Received by editor(s) in revised form: April 1, 2005
Published electronically: March 23, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society