Multiplicative Spectral Functionals On

Abstract

If $A$ is a commutative $C^{\star }$ -algebra and if $\unicode[STIX]{x1D719}:A\rightarrow \mathbb{C}$ is a continuous multiplicative functional such that $\unicode[STIX]{x1D719}(x)$ belongs to the spectrum of $x$ for each $x\in A$ , then $\unicode[STIX]{x1D719}$ is linear and hence a character of $A$ . This establishes a multiplicative Gleason–Kahane–Żelazko theorem for $C(X)$ .