Almost multiplicative functions and almost linear multiplicative functionals (Q1349038)

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 \(t\).