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Doctoral Thesis
Full name
Leandro Antunes
Knowledge Area
Date of Defense
São Paulo, 2019
Ferenczi, Valentin Raphael Henri (President)
Batista, Leandro Candido
Beanland, Kevin
Brech, Christina
Corrêa, Willian Hans Goes
Title in English
Light groups of isometries and polyhedrality of Banach spaces
Keywords in English
Combinatorial spaces
Distinguished points
Light groups
LUR renormings
Abstract in English
Megrelishvili defines light groups of isomorphisms of a Banach space as the groups on which the weak and strong operator topologies coincide and proves that every bounded group of isomorphisms of Banach spaces with the point of continuity property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, especially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X. We give an example of a Banach space X and an infinite countable group of isomorphisms G < GL(X) which is SOT-discrete but such that X does not admit a distinguished point for G, providing a negative answer to a question of Ferenczi and Rosendal. We also prove that every combinatorial Banach space is (V)- polyhedral. In particular, the Schreier spaces of countable order provide new solutions to a problem proposed by Lindenstrauss concerning the existence of an infinite-dimensional Banach space whose unit ball is the closed convex hull of its extreme points.
Title in Portuguese
Grupos leves de isometrias e poliedralidade de espaços de Banach
Keywords in Portuguese
Espaços combinatórios
Grupos leves
Pontos distintos
Renormações LUR
Abstract in Portuguese
Megrelishvili define grupos leves de isomorfismos de um espaço de Banach como os grupos em que as topologias fraca e forte do operador coincidem e prova que todo grupo limitado de isomorfismos de espaços de Banach com a propriedade do ponto de continuidade (PCP) é leve. Investigamos esse conceito para grupos de isomorfismos de espaços de Banach clássicos sem PCP, especialmente grupos de isometrias, e o relacionamos com a existência de renormações G-invariantes LUR ou uniformemente convexas. Damos um exemplo de um espaço de Banach X e um grupo enumerável infinito de isomorfismos G < GL(X) que é SOT-discreto mas tal que X não admite ponto distinto em relação a G, fornecendo uma resposta negativa a uma questão de Ferenczi e Rosendal. Também provamos que todos espaços de Banach combinatórios são (V)-poliedrais. Em particular, os espaços de Schreier de ordem enumerável fornecem novas soluções para um problema proposto por Lindenstrauss sobre a existência de um espaço de Banach de dimensão infinita cuja bola unitária seja igual a envoltória convexa fechada de seus pontos extremos.
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