Point multipliers and the Gleason–Kahane–Żelazko theorem

Abstract

Let $A$ be a Banach algebra, and let $\mathcal{X}$ be a left Banach $A$-module. In this paper, using the notation of point multipliers on left Banach modules, we introduce a certain type of spectrum for the elements of $\mathcal{X}$ and we also introduce a certain subset of $\mathcal{X}$ which behaves as the set of invertible elements of a commutative unital Banach algebra. Among other things, we use these sets to give some Gleason–Kahane–Żelazko theorems for left Banach $A$-modules.