A several variables Kowalski-S\lodkowski theorem for topological spaces

TL;DR

This paper generalizes the Kowalski-Słodkowski theorem to topological spaces and vector-valued functions, characterizing multiplicative functionals and operators in complex Banach algebras and Hardy spaces.

Contribution

It extends classical results to topological and vector-valued contexts, providing new characterizations of multiplicative functionals and operators.

Findings

Characterization of maps on $\mathcal{A}$-valued polynomials as compositions of multiplicative functionals and point evaluations.

Application of these characterizations to topological spaces of vector-valued functions.

Partial success in establishing a multiplicative GKZ theorem for Hardy spaces.

Abstract

In this paper, we provide a version of the classical result of Kowalski and S\l odkowski that generalizes the famous Gleason-Kahane-$\dot{\rm Z}$elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on $\mathcal{A}$-valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and S\l odkowski, as a composition of a multiplicative linear functional on $\mathcal{A}$ and a point evaluation on the polynomials, where $\mathcal{A}$ is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and S\l odkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces.