Banach Algebras of Vector-valued Functions
Azadeh Nikou, Anthony G. O'Farrell

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 -valued function algebra, a type of Banach algebra that consist of continuous -valued functions on some compact Hausdorff space, where 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 -valued function algebras. We give some specific examples.
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