A study in set-theoretic functional analysis: extensions of C_0(I)-valued operators on linearly ordered compacta and weaker forms of normality on Psi-spaces

• Autor(a) principal: Ronchim, Victor dos Santos
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-22112021-145510/
• topic: Análise funcional; Compactos totalmente ordenados; Extensão de operadores; Functional analysis; Linearly ordered compacta; Operator extension; Set-theoretic topology; Sobczyk's theorem; Teorema de Sobczyk; Topologia conjuntista

In the first part of this work we investigate generalizations of the classical theorem of Sobczyk, which states that every $c_0$-valued bounded operator defined on a closed subspace of a separable Banach space admits a bounded extension to the entire space. Towards this goal, we explore the generalization of this problem when $c_0$ is replaced by the non-separable space $c_0(I)$, addressing the problem of extending bounded operators defined on Banach unital subalgebras of $C(K)$, where $K$ is a linear compact space in an attempt to extend the results of D.V. Tausk and C. Correa. We describe a class of linear compact spaces, called separably determined, where the criteria for extending $c_0$-valued operators and the one for extending $c_0(I)$-valued operators are the same. On the second part of this work we examine weaker forms of normality in Mrówka-Isbell spaces. We study the concept of semi-normality in spaces providing structural results connecting normality, semi-normality and almost-normality. We define the separation concept of strongly $(\\aleph_0, <\\mathfrak c)$-separated almost disjoint families and prove the generic existence of completely separable strongly $(\\aleph_0, <\\mathfrak c)$-separated almost disjoint families under the assumption $\\mathfrak s=\\mathfrak c$ and $\\mathfrak b=\\mathfrak c$, answering a question from P. Szeptycki and S. Garcia-Balan.