Vector-valued spectra of Banach algebra valued continuous functions

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.