Banach Algebras of Vector-valued Functions

TL;DR

This paper introduces $E$-valued function algebras, a class of Banach algebras of continuous functions taking values in a Banach algebra $E$, and explores their boundary properties and examples.

Contribution

It defines and studies the properties of $E$-valued function algebras, extending classical function algebra concepts to vector-valued functions and analyzing their boundary structures.

Findings

Characterization of the Shilov boundary for $E$-valued function algebras

Identification of peak points in commutative $E$-valued algebras

Examples illustrating the structure of these algebras

Abstract

We introduce the concept of an $E$-valued function algebra, a type of Banach algebra that consist of continuous $E$-valued functions on some compact Hausdorff space, where $E$ is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative $E$-valued function algebras. We give some specific examples.