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Doctoral Thesis
Full name
André Santoleri Villa Barbeiro
Knowledge Area
Date of Defense
São Paulo, 2018
Fajardo, Rogerio Augusto dos Santos (President)
Aurichi, Leandro Fiorini
Franco Filho, Antonio de Padua
Kaufmann, Pedro Levit
Passos, Marcelo Dias
Title in Portuguese
Extensões conexas e espaços de Banach C(K) com poucos operadores
Keywords in Portuguese
Espaço C(K)
Espaço hereditariamente fracamente Koszmider
Espaço hereditariamente Koszmider
Extensão por funções contínuas
Poucos operadores
Princípio diamante
Problema de Efimov
Abstract in Portuguese
Este trabalho tem dois objetivos principais. Primeiramente, analisamos a preservação de conexidade na extensão de espaços compactos por funções contínuas, técnica utilizada por Koszmider para obter $C(K)$ indecomponível com poucos operadores. Mostramos que para todo compacto metrizável $K$ existe um desconexo $L$ que é obtido a partir de $K$ por uma quantidade finita de extensões por funções contínuas. Em seguida, enfatizamos a construção de espaços de Banach da forma $C(K)$ com poucos operadores, com a propriedade de que $C(L)$ tem poucos operadores, para todo fechado $L \subseteq K$. Assumindo o princípio diamante construímos uma família $(K_\xi)_{\xi < 2^{(2^\omega)}}$ de espaços conexos e hereditariamente Koszmider tais que todo operador de $C(K_\xi)$ em $C(K_\eta)$ é fracamente compacto, para $\xi$ diferente de $\eta$. Em particular, $(C(K_\xi))_{\xi < 2^{(2^\omega)}}$ é uma família de espaços de Banach indecomponíveis e dois a dois essencialmente incomparáveis, e cada espaço $K_\xi$ responde positivamente ao problema de Efimov. Apresentamos também um método de construção via forcing de um espaço compacto e conexo $K$ hereditariamente fracamente Koszmider.
Title in English
Connected extensions and Banach spaces C(K) with few operators
Keywords in English
C(K) space
Diamond principle
Efimov's problem
Extension by continuous functions
Few operators
Hereditarily Koszmider space
Hereditarily weakly Koszmider space
Abstract in English
This work has two main objectives. First, we analyze the preservation of connectedness in the extension of compact spaces by continuous functions, a technique used by Koszmider to obtain an indecomposable Banach space $C(K)$ with few operators. We show that for any metrizable compactum $K$ there exists a disconnected $L$ which is obtained from $K$ by finitely many extensions by continuous functions. Next, we emphasize the construction of Banach spaces of the form $C(K)$ with the property that $C(L)$ has few operators, for every closed $L \subseteq K$. Assuming the diamond principle we construct a family $(K_\xi)_{\xi < 2^{(2^\omega)}}$ of connected and hereditarily Koszmider spaces such that every operator from $C(K_\xi)$ into $C(K_\eta)$ is weakly compact, for $\xi$ different from $\eta$. In particular, $(C(K_\xi))_{\xi < 2^{(2^\omega)}}$ is a family of indecomposable and pairwise essentially incomparable Banach spaces, and each space $K_\xi$ responds positively to the Efimov's problem. We also present a method of construction using forcing of a compact and connected hereditarily weakly Koszmider space $K$.
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