On a connection between the fixed point property and \(L^{\infty}(\Omega,\Sigma,\mu)\)
In this article, the author investigates when weakly compact, convex subsets of \(L^\infty(\Omega, \Sigma, \mu)\) have the fixed point property for certain nonexpansive mappings. In particular, if \(K\) is a subset of \(L^\infty(\Omega, \Sigma, \mu)\), a mapping \(T:K\to K\) is called pointwise nonexpansive if, for every \(f, g \in K\), \(| T(f)(s) - T(g)(s)| \leq | f(s) - g(s)| \) for almost all \(s \in \Omega\). The author shows that weakly compact, convex subsets of \(L^\infty(\Omega, \Sigma, \mu)\) have the fixed point property for pointwise nonexpansive mappings. The author then introduces a family of Banach spaces of harmonic functions so that the Banach spaces continuously embed in \(L^\infty(\Omega)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) and the Banach spaces have the fixed point property for nonexpansive mappings. [Reviewer's remark: The author mistakenly writes (on pages 1722--1723) that ``\(\text{weak}^*\) compact subsets of \(L^\infty(\Omega, \Sigma, \mu)\) always have the fixed point property for nonexpansive self-mappings.'' In fact, \textit{D.\,Alspach} [Proc.\ Am.\ Math.\ Soc.\ 82, 423--424 (1981; Zbl 0468.47038)] provided an example of a weakly compact, convex subset of \(L^1[0,1]\) which fails the fixed point property for nonexpansive mappings. Since \(L^1[0,1]\) embeds isometrically in \(L^\infty[0,1]\), Alspach's example gives a weakly (and \(\text{weak}^*\)) compact, convex set in \(L^\infty[0,1]\) without the fixed point property for nonexpansive self-mappings.]
scientific article; zbMATH DE number 5142225
scientific article