Light groups of isometries and polyhedrality of Banach spaces

• Autor(a) principal: Antunes, Leandro
• Data de Publicação: 2019
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-26072019-151133/
• topic: Combinatorial spaces; Distinguished points; Espaços combinatórios; Grupos leves; Light groups; LUR renormings; Poliedralidade; Polyhedrality; Pontos distintos; Renormações LUR

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.