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

 

A note on operators fixing cotype subspaces of $ C[0,1]$


Author: I. Gasparis
Journal: Proc. Amer. Math. Soc. 142 (2014), 1633-1639
MSC (2010): Primary 46B03; Secondary 47B38
Published electronically: February 10, 2014
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Abstract: Let $ K$ be a compact, metrizable space. Let $ X$ be a closed, linear subspace of $ C(K)$ spanned by a normalized weakly null sequence $ (f_n)$ such that $ (\vert f_n\vert)$ satisfies a lower $ q$ estimate on disjoint blocks with positive coefficients for some $ 1 < q < \infty $. It is proved that every $ w^*$-compact subset of $ B_{C(K)^*}$ which norms $ X$ is non-separable in norm. This provides an alternative proof of Bourgain's result that every $ w^*$-compact subset of $ B_{C(K)^*}$ which norms a subspace with non-trivial cotype is non-separable in norm.


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Additional Information

I. Gasparis
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece
Address at time of publication: Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Heroon Polytechneiou 9, Athens, 15780, Greece
Email: ioagaspa@math.ntua.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-11913-7
PII: S 0002-9939(2014)11913-7
Keywords: Operators on spaces of continuous functions, lower \(q\) estimates
Received by editor(s): January 9, 2012
Received by editor(s) in revised form: April 5, 2012, and June 5, 2012
Published electronically: February 10, 2014
Additional Notes: This research was partially supported by grant ARISTEIA 1082
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.