Almost multiplicative linear functionals and entire functions

Abstract

Let $T$ be a unital, continuous linear functional defined on complex Banach algebra $A$. First, we prove an approximate version of the Gleason-Kahane-\.Zelazko theorem: given $\epsilon >0$, there exists an $M>0$ such that, if $$ T(\exp x)\neq 0,\quad x\in A,\ \|x\|\leq M, $$ then $T$ is $\epsilon $-almost multiplicative. Then, we show that this result remains true if the exponential function is replaced by a nonsurjective entire function~$F$ with $F'(0)\neq 0$.