Point multipliers and the Gleason–Kahane–Żelazko theorem

Abstract

Let A be a Banach algebra, and let 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 X and we also introduce a certain subset of 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.