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Doctoral Thesis
Full name
Willian Hans Goes Corrêa
Knowledge Area
Date of Defense
São Paulo, 2018
Ferenczi, Valentin Raphael Henri (President)
Brech, Christina
Carrera, Wilson Albeiro Cuellar
Castillo, Jesús Maria Trinidad Fernandez
Pellegrino, Daniel Marinho
Title in English
Results on twisted sums of Banach and operator spaces
Keywords in English
Complex interpolation
Operator spaces
Twisted sums
Abstract in English
In this work we study twisted sums induced by complex interpolation, of Banach spaces as well as of operator spaces. In the first part of the thesis we focus on Banach spaces, and clarify how interpolation of families, as of couples, induces an extension of the interpolation space, called the derived space. We study how the types and cotypes of the spaces being interpolated determine the triviality or singularity of the derived space, and we apply the results to the study of submodules of the Schatten classes and in the obtainment of nontrivial twisted sums in which all of the three spaces in the short exact sequence do not have the approximation property. In the second part we develop the theory of twisted sums in the category of operator spaces and present many examples of twisted sums which are completely singular and completely nontrivial. In particular, we solve two versions of Palais' problem for operator spaces.
Title in Portuguese
Resultados de somas torcidas de espaços de Banach e espaços de operadores
Keywords in Portuguese
Espaços de operadores
Interpolação complexa
Somas torcidas
Abstract in Portuguese
Neste trabalho estudamos somas torcidas induzidas por interpolação complexa, tanto de espaços de Banach como de espaços de operadores. Na primeira parte da tese focamos em espaços de Banach, e esclarecemos como a interpolação de famílias, assim como a de pares, gera uma extensão do espaço interpolado, chamada de espaço derivado. Estudamos como os tipos e cotipos dos espaços sendo interpolados influenciam na trivialidade ou singularidade do espaço derivado, e aplicamos os resultados para o estudo de submódulos das classes de Schatten e para a obtenção de somas torcidas não-triviais em que os três espaços da sequência exata curta não possuem a propriedade da aproximação. Na segunda parte, desenvolvemos a teoria de somas torcidas na categoria de espaços de operadores, e apresentamos vários exemplos de somas torcidas completamente singulares e completamente não-triviais nessa categoria. Em particular, resolvemos duas versões do problema de Palais para espaços de operadores.
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