Banach spaces with various complex structures

Author(s)

Wilson Albeiro Cuellar Carrera

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2015-04-29

Examining board members

Valentin Raphael Henri Ferenczi; Jorge Lopez Abad; Eloi Medina Galego; Manuel González Ortiz; Yolanda Moreno Salguero

• Advisor: Valentin Raphael Henri Ferenczi

Abstract

In this work, we study some aspects of the theory of complex structures in Banach spaces. We show that if a real Banach space $X$ has the property $P$, then all its complex structures also satisfy $P$, where $P$ is any of the following properties: bounded approximation property, \\emph{G.L-l.u.st}, being injective and being complemented in a dual space. We address the problem of uniqueness of complex structures in Banach spaces with subsymmetric basis by proving that a real Banach space $E$ with subsymmetric basis and isomorphic to the space of sequences $E [E]$ admits a unique complex structure. On the other hand, we show an example of Banach space with exactly $\\omega$ different complex structures. We also use the theory of complex structures to study the classical problem of hyperplanes in the Kalton-Peck space $Z_2$. In order to distinguish between $Z_2$ and its hyperplanes we wonder whether the hyperplanes admit complex structures. In this sense we prove that no complex structure on $\\ell_2$ can be extended to a complex structure on the hyperplanes of $Z_2$ containing the canonical copy $l_2$. We also constructed a complex structure on $l_2$ that can not be extended to any operator in $Z_2$. (AU)

• FAPESP's process: 10/17512-6 - Banach spaces with various complex structures
• Grantee: Wilson Albeiro Cuellar Carrera
• Support Opportunities: Scholarships in Brazil - Doctorate