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 $C(X)$.

中文翻译：

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多重谱函数

如果$澳元是可交换的$C^{\星}$-代数和如果$\unicode[STIX]{x1D719}:A\rightarrow \mathbb{C}$是一个连续的乘法泛函，使得$\unicode[STIX]{x1D719}(x)$属于谱系$x$对于每个$x\in 澳元， 然后$\unicode[STIX]{x1D719}$是线性的，因此具有$澳元. 这建立了一个乘法 Gleason-Kahane-Żelazko 定理$C(X)$.