Almost multiplicative functions and almost linear multiplicative functionals
type: Item
instance of: scholarly article
title: Almost multiplicative functions and almost linear multiplicative functionals
zbMATH Open document ID: 1007.39024
author: Peter Šemrl
published in: Aequationes Mathematicae
publication date: 2002-05-21T00:00:00Z
review text: Among others the following result is offered. If \(\delta>0\), \(A\) is a real Banach algebra, and \(\varphi: A\to\mathbb{R}\) a \(\delta\)-homomorphism, that is, \[ |\varphi(x+y)-\varphi(x)-\varphi(y)|\leq\delta(\|x\|+\|y\|),\quad |\varphi(xy)-\varphi(x)\varphi(y)|\leq \delta\|x\|\|y\|, \] then \(2\sup_{x\in A\setminus \{0\}} |\varphi(x)|/\|x\|\leq 1+(1+4\delta)^{1/2}.\) The last inequality is sharp. The proof proceeds through a sharpening of a result of \textit{J. A. Baker} [Proc. Am. Math. Soc. 80, 411-416 (1980; Zbl 0448.39003)] that, under the added assumption \(\delta<1/5,\) shows that an \(f:\mathbb{R}\setminus{0}\to\mathbb{R}\) satisfies \(|f(st)-f(s)f(t)|\leq\delta\) \((ts\neq 0)\) if it is multiplicative or `close' to the constants 1 or 0 or to sign \(t\).
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B72
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 1742897
zbMATH Keywords: functional equations
zbMATH Keywords: stability
zbMATH Keywords: approximate homomorphisms
zbMATH Keywords: real Banach algebra
zbMATH Keywords: almost multiplicative functions
zbMATH Keywords: almost linear multiplicative functionals
MaRDI profile type: Publication
full work available at URL: s00010-002-8016-6
OpenAlex ID: W2025793054
DOI: 10.1007/S00010-002-8016-6
description: scientific article; zbMATH DE number 1742897