Immersions and isomorphisms between spaces of continuous functions

• Grant number: 25/00001-4
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: June 01, 2025
• End date: May 31, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Eloi Medina Galego
• Grantee: Anna Júlia Gonçalves Veronezi
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/12916-1 - Geometry of Banach spaces, AP.TEM

Abstract

Let \(K\) be a locally compact Hausdorff space and let \(X\) be a Banach space. By \(C_0(K, X)\) we denote the Banach space of all \(X\)-valued continuous functions on \(K\) that vanish at infinity, equipped with the supremum norm. When \(K\) is compact, we write \(C(K, X)\). In this context, the spaces \(C(K)\), where \(X\) is the scalar field, play a central role in the theory of Banach spaces. This project aims to study the linear and nonlinear geometry of \(C_0(K, X)\) spaces, with an emphasis on generalizations of the Banach-Stone theorem. The proposed study is part of the FAPESP thematic project 23/12916-1: Geometry of Banach Spaces. Our project is divided into five parts.We begin by remembering that in the article "Into isomorphisms of spaces of continuous functions" (1984), Jarosz showed that if there exists a linear isomorphism \(T\) from \(C(K)\) to \(C_0(S)\) such that \(\|T\|\|T^{-1}\| < 2\), then there exists a closed subset \(S_0 \subseteq S\) and a surjective continuous function \(\varphi: S_0 \to K\). As a consequence, we have the following geometric conclusion: (*) the operator \(\Phi\), defined by \(\Phi(f)(s) = f(\varphi(s))\), is a linear isometry from \(C(K)\) into \(C(S_0)\).In Part 1, we will investigate whether it is possible to generalize Jarosz's result to the case where \(K\) is not compact, analyzing whether the geometric conclusion (*) still holds. This problem is related to Problem 1.7 left open by Galego and da Silva in the article "On the structure of into isomorphisms between spaces of continuous functions" (2023).Moreover, in 2018, Galego and Rincón-Villamizar, in the article "Continuous maps induced by embeddings of \(C(K)\) spaces into \(C_0(S, X)\) spaces", showed that if \(T\) is a linear isomorphism from \(C_0(K)\) to \(C_0(S, X)\) with \(\|T\|\|T^{-1}\| < S(X)\) (Schäffer's constant), then there exists a locally compact subset \(S_0 \subseteq S\) and a continuous function \(\varphi: S_0 \to K\). In Part 2, inspired by the above mentioned result, we will study the problem of finding conditions so that into linear isomorphisms \(C_0(K)\) to \(C_0(S, X)\) also produce the geometric conclusion (*) of Jarosz's theorem.To continue, in 2022, Galego and da Silva, in the article "On \(C_0(S, X)\)-distortion of the class of all separable Banach spaces", showed that if there is an \((M, L)\)-quasi-isometry from \(C_0(K)\) to \(C_0(S, X)\) with \(M^2 < S(X)\), then \(K\) is the continuous image of a closed subset of \(S\). So, in part 3, we will look for the \((M, L)\)-quasi-isometry from \(C_0(K)\) to \(C_0(S, X)\) that eventually produces the conclusion of Jarosz's theorem.Next, in 2019, Galego and da Silva, in the article "Isomorphisms of \(C_0(K, X)\) spaces with large distortion", demonstrated that if \(X\) is strictly convex, of finite dimension greater than 2, \(K\) and \(S\) are locally compact, and \(T\) is a linear isomorphism from \(C_0(K, X)\) to \(C_0(S, X)\) satisfying \(\|T\|\|T^{-1}\| \leq S(X)\), then \(K\) and \(S\) are homeomorphic. Furthermore, in 2024, Galego showed in "A stronger form of Banach-Stone theorem to \(C_0(K, X)\) spaces including the cases \(X=\ell_p^2, 1