Almost multiplicative linear functionals and entire functions (Q521282)

A linear functional \(f\) on a Banach algebra \(A\) is said to be \(\epsilon\)-almost multiplicative if \(|f(xy)-f(x)f(y)|\leq\epsilon\) for all \(x,y\in A\) with \(\|x\|=\|y\|=1\). Let \(F\) be a non-surjective entire function with \(F'(0)\not=0 \) and \(f\) a continuous linear functional on a Banach algebra \(A\) with unit \(e\), such that \(f(e)=1\). The main result of the paper states that, for a given \(\epsilon>0\), there exists an \(M>0\) depending only upon \(\epsilon\) and \(\|f\|\) such that, if \(f(F(x))\not=0\) for \(\|x\|\leq M\), then \(f\) is \(\epsilon\)-almost multiplicative.