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Doctoral Thesis
Full name
Rogerio Augusto dos Santos Fajardo
Knowledge Area
Date of Defense
São Paulo, 2007
Galego, Eloi Medina (President)
Ascui, Jorge Tulio Mujica
Bianconi, Ricardo
Ferenczi, Valentin
Matos, Mario Carvalho de
Title in Portuguese
Construções consistentes de espaços de Banach C (K) com poucos operadores
Keywords in Portuguese
espaços de Banach
espaços indecomponíveis
Abstract in Portuguese
Neste trabalho aplicamos técnicas de combinatória infinitária e forcing na teoria dos espaços de Banach, investigando propriedades dos espaços de Banach da forma C(K), formado pelas funções reais contínuas sobre K com a norma do supremo, com poucos operadores, no sentido de que todo operador em C(K) é da forma gI+S, onde I é o operador identidade, g pertence a C(K) e S é fracamente compacto. Enfatizamos as construções onde K é conexo, o que implica que C(K) é indecomponível. Assumindo Axioma Diamante, um axioma combinatório mais forte que a Hipótese do Contínuo, construímos um espaço de Banach C(K) tal que C(L) tem poucos operadores, para todo L subespaço fechado de K. Sob a Hipótese do Contínuo construímos um espaço C(K) indecomponível com poucos operadores tal que K contém $\beta N$ homeomorficamente. Em ZFC construímos um espaço C(K) com poucos operadores em um sentido estritamente mais fraco. Também mostramos a existência de pelo menos contínuo espaços de Banach C(K) indecomponíveis dois a dois essencialmente incomparáveis. Usando forcing provamos que existe consistentemente um espaço de Banach C(K) de densidade menor que contínuo com poucos operadores e um C(K) indecomponível de densidade menor que contínuo.
Title in English
Consistent constructions of Banach spaces C(K) with few operators
Keywords in English
Banach spaces
indecomposable Banach spaces
Abstract in English
In this work we apply techniques of infinitary combinatorics and forcing in Banach spaces theory, investigating the compact topological spaces K such that the Banach space C(K), consisting of the continuous real-valued functions on K with the supremum norm, has few operators, in the sense that all operators on C(K) have the form gI+S, where I is the identity operator, g\ belongs to C(K) and S is weakly compact. We emphasize the constructions where K is connected, which implies that C(K) is indecomposable. Assuming Diamond Axiom, a combinatoric axiom stronger than the continuum hypothesis, we construct a Banach space C(K) where C(L) has few operators, for every L closed subspace of K. Under continuum hypothesis we construct an indecomposable C(K) with few operators such that K contains $\beta \mathbb$ homeomorphically. In ZFC we construct a space C(K) with few operators in a strictly weaker sense. We also show the existence of at least continuum pairwise essentially incomparable indecomposable Banach spaces C(K). Using forcing, we prove that there exists consistently a Banach space C(K) of density smaller than continuum having few operators and an indecomposable C(K) of density smaller than continuum.
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