A weaker Gleason–Kahane–Żelazko theorem for modules and applications to Hardy spaces
Volume 164 / 2021
Colloquium Mathematicum 164 (2021), 273-282
MSC: Primary 46H05; Secondary 30H10, 46H25.
DOI: 10.4064/cm8015-9-2019
Published online: 5 October 2020
Abstract
Let $A$ be a complex unital Banach algebra and $M$ be a left $A$-module. Let $\Lambda :M\rightarrow \mathbb {C}$ be a map that is not necessarily linear. We establish conditions for $\Lambda $ to be linear and of multiplicative kind, from its behavior on a small subset of $M$. We do not assume $\Lambda $ to be continuous throughout. As an application, we give a characterization of weighted composition operators on the Hardy space $H^\infty $.
