Coarse versions of the classical Banach-Stone theorem

• Grant number: 12/21091-1
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2013
• End date: February 28, 2015
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Eloi Medina Galego
• Grantee: André Luis Porto da Silva
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The candidate will study two recent papers (2011) written by RafaB Górak, "Coarse version of the Banach-Stone theorem" and "Pertubations of isometries between Banach spaces".The first one presents an improvement of a result due to Y. Dutrieux and N. J. Kalton, similar to Banach-Stone theorem which includes bi-Lipschitz mappings. An abstract of this paper is as follows.Let $X$ be a compact space, and $C(X)$ the Banach space of all real-valued continuous functions in $X$. A $(\varepsilon,\delta)$-net $N_E$ in a Banach space $E$ is a subset $N$ of $E$ with the following properties:(i) $d(x,y)\geq\delta$ for any two distinct elements $x,y,\in N$.(ii)$d(x,N)<\varepsilon$ for all $x\in E$.For a Lipschitz mapping $T$ from a net in the Banach Space $E$ to another Banach space $F$, we denote by $l(T)$, the Lipschitz norm of T, and we put$$d_N(E,F) = inf\{l(T)\cdot l(T^{-1})\},$$where the infimum is taken over all bi-Lipschitz maps $T$ between the nets $N_E$ and $N_F$ in the Banach spaces $E$ and $F$, respectively. Among many other things, Y. Dutriex and N. J. Kalton showed that, for two compact spaces $X$ and $Y$, if $d_N(E,F)<17/16$, then the spaces X and Y are homeomorfic. Through a sequence of estimates,the author of the paper further shows the the conclusion mentioned above is valid again if $d_N(E,F)<6/5$. The author also estimates the distance of a coarse quasi-isometry T of the Banach spaces C(X) and C(Y) to an isometry of these spaces. To conclude this paper, the author proposes the two following problems:Problem 1. Is it true that for all compact spaces $X$ and $Y$ the inequality $d_N(C(X),C(Y))2$ implies that $X$ and $Y$ are homeomorphic?Problem 2. What is the answer to Problem 4.1 if we consider only countable compacts $X$ and $Y$? What if $X$ is a convergente sequence?Finally, the second paper presents the same result for $C_0(X)$ which $X$ is locally compact.

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

SILVA, André Luis Porto da. Coarse versions of the classical Banach-Stone theorem. 2015. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) São Paulo.