Homomorphisms on function lattices
type: Item
instance of: scholarly article
title: Homomorphisms on function lattices
zbMATH Open document ID: 1058.54008
author: Jesús A. Jaramillo
author: M. I. Garrido
published in: Monatshefte für Mathematik
publication date: 2004-09-22T00:00:00Z
review text: Let \(X\) be a completely regular space, \(C(X)\) the set of all continuous functions from \(X\) into the real numbers \(\mathbb R\), and \(L\subset C(X)\) a unital vector lattice. The structural space \(H(L)\) is the set of all lattice homomorphisms from \(L\) into \(\mathbb R\). For each \(x\in X\) the point evaluation map \(\delta_x\) belongs to \(H(L)\), where \(\delta_x(f)=f(x)\) for each \(f\in L\). If \(L\) separates points and closed sets, then \(X\) is a topolopical subspace of \(H(L)\), and \(H(L)\) is a realcompactification of \(X\). Indeed, all realcompactifications of \(X\) have the form \(H(L)\) for some \(L\subset C(X)\). In the present paper, the authors study the realcompactifications \(H(L)\) and compactifications \(H(L^*)\) of \(X\), where \(L^*\) denotes the set of bounded elements in \(L\), and \(H(C^* (X))\) is the Stone-Čech compactification \(\beta X\) of \(X\). They call \(X\) \(L\)-realcompact in case \(H(L)=X\). They show, among other things, that for each \(\varphi\in H(L)\) there exists \(\xi\in\beta X\) such that, for all \(f\in L\), \(f^\beta(\xi)\neq\infty\) and \(\varphi(f)=f^\beta(\xi)\). They study the case where \(L=\text{Lip}(X)\) is the set of real valued Lipschitz functions on a metric space \(X\) and show that \(X\) is \(L\)-realcompact if, and only if, every closed ball in \(X\) is compact. Other results concern a vector lattice isomorphism \(T\) between \(\text{Lip}(X)\) and \(\text{Lip} (Y)\) which maps unit element to unit element. They show that if \(X,Y\) are complete, then \(T\) exists if, and only if, \(X\) and \(Y\) are Lipschitz homeomorphic. A similar result for \(\text{Lip}^*(X)\) and \(\text{Lip}^* (Y)\) was obtained by \textit{N. Weaver} [Pac. J. Math. 164, No. 1, 179--193 (1994; Zbl 0797.46007)].
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 46E05
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D60
zbMATH DE Number: 2102088
zbMATH Keywords: lattice homomorphism
zbMATH Keywords: Lipschitz structure
zbMATH Keywords: Banach-Stone theorem
zbMATH Keywords: unital vector lattice
MaRDI profile type: Publication
full work available at URL: s00605-002-0011-4
OpenAlex ID: W1987563602
DOI: 10.1007/S00605-002-0011-4
Recommended article: Homomorphisms on lattices of continuous functions
Recommended article: Q3147503
Recommended article: Q3728044
Recommended article: Q5189288
Recommended article: Homomorphisms of the continuous lattices
Recommended article: Q4513513
Recommended article: Homomorphisms on Function Algebras
Recommended article: Q4763071
Recommended article: Homomorphisms on some function algebras
Recommended article: Function lattices and compactifications
description: scientific article; zbMATH DE number 2102088