Stability of subspaces, complemented subspaces and isomorphisms in Banach Spaces

• Grant number: 14/08176-3
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: August 01, 2014
• End date: June 30, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
• 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

Our goal is to study and extend results obtained by Saab-Saab, Cembranos-Mendoza and Khmyleva regarding complemented subspaces and isomorphisms in Banach spaces. Our main goal is to generalize the following result proved by Saab-Saab: Theorem 1: Let $K$ be a compact Hausdorff space and $X$ be a Banach space. Then $C(K.X)$ contains a complemented copy of $\ell_1$ if, and only if, $X$ contains a complemented copy of $\ell_1$.Specifically, we are interested in obtaining a stronger version of Theorem 1, replacing $\ell_1$ with $\ell_1(\Gamma)$, where $\Gamma$ is an infinite set, or with $L_1[0,1]$.Another problem we will study is related to the following result, proved independently by Cembranos-Mendoza and Khmyleva: Theorem 2: The Banach spaces $c_0(\mathbb{N},\ell_\infty)$ and $\ell_\infty(\mathbb{N},c_0)$ are not isomorphic. As a consequence of this result, the Banach spaces $\ell_\infty(\mathbb{N},c_0)$ and $C(\beta \mathbb{N}, c_0)$ are not isomorphic, where $\beta \mathbb{N}$ is the Stone-ech compactification of the set of the natural numbers. It is well known that the Banach spaces $\ell_\infty(\mathbb{N},X)$ and $C(\beta \mathbb{N}, X)$ are isomorphic if $X$ is finite-dimensional. This suggests the following open problem: Problem: Let $X$ be a Banach space such that the Banach spaces $\ell_\infty(\mathbb{N},X)$ and $C(\beta \mathbb{N}, X)$ are isomorphic. Must $X$ be finite-dimensional? (AU)

Academic Publications

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

CORTES, Vinicius Morelli. Geometrical aspects of Co(K,X) spaces. 2017. Doctoral Thesis - Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) São Paulo.