Copies of $c_{0}(\Gamma )$ in $C(K, X)$ spaces
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- by Elói Medina Galego and James N. Hagler PDF
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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:
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[(1)] If $C([0,1], X)$ contains a copy of $c_0(\aleph _1)$, then $X$ contains a copy of $c_0(\aleph _1)$.
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[(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
- MR Author ID: 647154
- Email: eloi@ime.usp.br
- James N. Hagler
- Affiliation: Department of Mathematics, University of Denver, Denver, Colorado 80208
- Email: jhagler@math.du.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3843-3852
- MSC (2010): Primary 46B03; Secondary 46B25
- DOI: https://doi.org/10.1090/S0002-9939-2012-11208-0
- MathSciNet review: 2944725