Vector-valued spectra of Banach algebra valued continuous functions

Abstract

Let E be a commutative unital Banach algebra, X a compact Hausdorff space, and \( \mathcal {A} \subset C(X,E)\) a Banach E-valued function algebra. An E-valued spectrum \(E{\text {-}}{\textsc {sp}}(f)\) of every \(f\in \mathcal {A} \) is introduced and investigated, and it is shown that \(E{\text {-}}{\textsc {sp}}(f)\) can be determined by E-valued characters of \( \mathcal {A} \). For the so-called natural E-valued function algebras, such as C(X, E) and \({{\mathrm{Lip}}}(X,E)\), we get \(E{\text {-}}{\textsc {sp}}(f)=f(X)\). When \(E = \mathbb {C}\), E-valued characters reduce to characters and E-valued spectra reduce to classical spectra.