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

 

Copies of $ c_{0}(\Gamma)$ in $ C(K, X)$ spaces


Authors: Elói Medina Galego and James N. Hagler
Journal: Proc. Amer. Math. Soc. 140 (2012), 3843-3852
MSC (2010): Primary 46B03; Secondary 46B25
Published electronically: March 2, 2012
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Abstract: We extend some results of Rosenthal, Cembranos, Freniche,
E. Saab-P. Saab and Ryan to study the geometry of copies and complemented copies of $ c_{0}(\Gamma )$ in the classical Banach spaces $ C(K, X)$ in terms of the cardinality of the set $ \Gamma $, of the density and caliber of $ K$ and of the geometry of $ X$ and its dual space $ X^*$. Here are two sample consequences of our results:

  1. If $ C([0,1], X)$ contains a copy of $ c_0(\aleph _1)$, then $ X$ contains a copy of $ c_0(\aleph _1)$.
  2. $ C(\beta \mathbb{N},X)$ contains a complemented copy of $ c_{0}(\aleph _{1})$ if and only if $ X$ contains a copy of $ c_{0}(\aleph _{1})$.
Some of our results depend on set-theoretic assumptions. For example, we prove that it is relatively consistent with ZFC that if $ C(K)$ contains a copy of $ c_0(\aleph _1)$ and $ X$ has dimension $ \aleph _1$, then $ C(K,X)$ contains a complemented copy of $ c_0(\aleph _1)$.

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Additional Information

Elói Medina Galego
Affiliation: Department of Mathematics, University of São Paulo, São Paulo, Brazil 05508-090
Email: eloi@ime.usp.br

James N. Hagler
Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
Email: jhagler@math.du.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11208-0
PII: S 0002-9939(2012)11208-0
Keywords: $c_0(\gamma)$ spaces, $C(K,X)$ spaces, Josefson-Nissenzweig-$\alpha$ ($JN_\alpha)$ property.
Received by editor(s): February 25, 2011
Received by editor(s) in revised form: April 4, 2011, and April 21, 2011
Published electronically: March 2, 2012
Additional Notes: The authors thank the referee for insightful and helpful comments, which have led to a significant improvement in the exposition.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.