Continuous multiplicative spectral functionals on Hermitian Banach algebras

Let \(\mathfrak {A}\) $\mathfrak{A}$ be a unital Hermitian Banach algebra with the spectrum of \(a\in \mathfrak {A}\) $a\in \mathfrak{A}$ denoted by \(\sigma _\mathfrak {A}(a)\) ${\sigma }_{\mathfrak{A}}(a)$ . We show that if a continuous and multiplicative function \(\phi : \mathfrak {A}\rightarrow \mathbb {C}\) $\varphi :\mathfrak{A}\to \mathbb{C}$ satisfies \(\phi (a)\in \sigma (a)\) $\varphi (a)\in \sigma (a)$ for all \(a\in \mathfrak {A}\) $a\in \mathfrak{A}$ , then \(\phi \) $\varphi$ is linear and hence a character of \(\mathfrak {A}\) $\mathfrak{A}$ .