Geometrical aspects of Co(K,X) spaces

Author(s)

Vinícius Morelli Cortes

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2017-06-27

Examining board members

Eloi Medina Galego; Leandro Candido Batista; Pedro Levit Kaufmann; Antonio Roberto da Silva; Daniela Mariz Silva Vieira

• Advisor: Eloi Medina Galego

Abstract

The goal of this work is two-fold. First, we study the complemented copies of co(T) in Banach spaces, where T is an innite cardinal. We extend to the uncountable case a classical result by T. Schulmprecht that characterizes the complemented copies of co in a Banach space X. We use this new characterization to extend results by G. Emmanuele, F. Bombal, D. Leung and F. Räbiger concerning the complemented copies of co in the classical Banach spaces `p(I,X), where p T[1, &#8734 ] and I is a non-empty set. We also obtain a new result involving the complemented copies of co(T) in Co(K,X) spaces, where Kis a locally compact Hausdor space. Next, we turn our attention to a vector-valued extension of the classical Banach-Stone theorem obtained by K. Jarosz. Studying several constants introduced by R. James, J. Schäffer, M. Baronti, E. Casini and P. Pappini, we obtain a new relationship between the moduli of convexity of Xand X*, which has independent interest. We then apply this relationship to prove a new X-valued generalization of the Banach-Stone theorem that simultaneously extends the aforementioned result by Jarosz and also shows that this result is, in fact, a consequence of a theorem obtained recently by F. Cidral, E. Galego and M. Rincón-Villamizar. (AU)

• FAPESP's process: 14/08176-3 - Stability of subspaces, complemented subspaces and isomorphisms in Banach Spaces
• Grantee: Vinicius Morelli Cortes
• Support Opportunities: Scholarships in Brazil - Doctorate