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Doctoral Thesis
Full name
Wilson Albeiro Cuellar Carrera
Knowledge Area
Date of Defense
São Paulo, 2015
Ferenczi, Valentin Raphael Henri (President)
Abad, Jorge Lopez
Galego, Eloi Medina
Ortiz, Manuel González
Salguero, Yolanda Moreno
Title in Portuguese
Espaços de Banach com várias estruturas complexas
Keywords in Portuguese
Bases subsimétricas
Espaço de Kalton-Peck
Espaços com `poucos operadores'
Estruturas complexas
Somas torcidas
Abstract in Portuguese
No presente trabalho, estudamos alguns aspectos da teoria de estruturas complexas em espaços de Banach. Demonstramos que se um espaço de Banach real $X$ tem a propriedade $P$, então todas as estruturas complexas em $X$ também satisfazem $P$, quando $P$ é qualquer uma das seguintes propriedades: propriedade de aproximação limitada, \emph{G.L-l.u.st}, ser injetivo e ser complementado num espaço dual. Abordamos o problema da unicidade de estruturas complexas em espaços de Banach com base subsimétrica, provando que um espaço de Banach real $E$ com base subsimétrica e isomorfo ao espaço de sequências $E[E]$ admite estrutura complexa única. Por outro lado, apresentamos um exemplo de espaço de Banach com exatamente $\omega$ estruturas complexas distintas. Também usamos a teoria de estruturas complexas para estudar o clássico problema dos hiperplanos no espaço $Z_2$ de Kalton-Peck. Com o propósito de distinguir $Z_2$ de seus hiperplanos nos perguntamos se os hiperplanos admitem estrutura complexa. Nesse sentido, provamos que os hiperplanos de $Z_2$ contendo a cópia canônica de $\ell_2$ não admitem estruturas complexas que sejam extensões de estruturas complexas em $\ell_2$. Também construímos uma estrutura complexa em $\ell_2$ que não pode-se estender a nenhum operador em $Z_2$.
Title in English
Banach spaces with various complex structures
Keywords in English
Complex structures
Kalton-Peck space
Spaces with `few operators'
Subsymmetric basis
Twisted sums
Abstract in English
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 $Z_2$.
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