Linear maps determining the norm topology

Abstract

Let A A be a Banach function algebra on a compact space X X , and let a ∈ A a\in A be such that for any scalar λ \lambda the element a + λ e a+\lambda e is not a divisor of zero. We show that any complete norm topology on A A that makes the multiplication by a a continuous is automatically equivalent to the original norm topology of A A . Related results for general Banach spaces are also discussed.