Vector-valued spectra of Banach algebra valued continuous functions

TL;DR

This paper explores the vector-valued spectrum of functions in Banach algebra valued function algebras, establishing its properties and relation to $A$-characters, and shows that for natural algebras it coincides with the function's image.

Contribution

It defines the vector-valued spectrum for $A$-valued functions and proves its characterization via $A$-characters, extending classical spectral theory to vector-valued contexts.

Findings

The $A$-valued spectrum contains the function's image.

For natural $A$-valued algebras, the spectrum equals the function's image.

When $A=\mathbb{C}$, the theory reduces to classical spectral results.

Abstract

Given a compact space $X$, a commutative Banach algebra $A$, and an $A$-valued function algebra $\mathscr{A}$ on $X$, the notions of vector-valued spectrum of functions $f\in\mathscr{A}$ are discussed. The $A$-valued spectrum $\vec{SP}_A(f)$ of every $f\in\mathscr{A}$ is defined in such a way that $f(X) \subset \vec{SP}_A(f)$. Utilizing the $A$-characters introduced in (M. Abtahi, \textit{Vector-valued characters on vector-valued function algebras}, \texttt{arXiv:1509.09215 [math.FA]}), it is proved that $\vec{SP}_A(f) = \{\Psi(f):\text{$\Psi$ is an $A$-character of $\mathscr{A}$}\}$. For the so-called natural $A$-valued function algebras, such as $C(X,A)$ and $Lip(X,A)$, we see that $\vec{SP}_A(f)=f(X)$. When $A = \mathbb{C}$, Banach $A$-valued function algebras reduce to Banach function algebras, $A$-characters reduce to characters, and $A$-valued spectrums reduce to usual spectrums.