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

• Grant number: 11/15567-0
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2012
• End date: February 28, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Eloi Medina Galego
• Grantee: Vinicius Morelli Cortes
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The candidate will study in detail a recent paper by his advisor that will be published in 2012 in Proc. Amer. Math. Soc. This paper suggests many lines of research associating Banach spaces to set theory. In two years, we will try to advance in the study of the geometry of the subspaces of $C(K, X)$ that are isomorphic to some $c_{0}(\gamma)$. Below is the abstract of this paper."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 theclassical Banach spaces $C(K, X)$ in terms of the cardinality of theset $\Gamma$, of the density and caliber of $K$ and of thegeometry of $X$ and its dual space $X^*$. Here are two sample consequences of our results: \begin{enumerate}\item[(1)] If $C([0,1], X)$ contains a copy of $c_0(\aleph_1)$, then $X$ contains a copy of $c_0(\aleph_1)$.\end{enumerate}\begin{enumerate}\item[(2)] $C(\beta \mathbb N,X)$contains a complemented copy of $c_{0}(\aleph_{1})$ if and only if $X$contains a copy of $c_{0}(\aleph_{1})$.\end{enumerate}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 $c_0(\aleph_1)$".

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

CORTES, Vinicius Morelli. Cópias de c0(T) em espaços C(K,X). 2014. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) São Paulo.