The theorem of Gleason-Kahane-Zelazko in a commutative symmetric Banach algebra (Q1071270)

In [Stud. Math. 29, 339-343 (1968; Zbl 0155.458)] \textit{J.-P. Kahane} and \textit{W. Żelazko} show that the theorem of Gleason-Kahane-Zelazko implies the following analogous characterization of multiplicative complex measures: a complex regular Borel measure \(\mu\) on a compact Hausdorff space X, which possesses the mean value property \[ \int_{X}f d\mu \in image(f) \] for all f in \(C_{{\mathbb{C}}}(X)\) is just a one point measure. Now in this paper we take just the opposite approach: at first the measure theorem is proved. In contrast to the known proofs of the theorem of Gleason-Kahane-Zelazko [see \textit{A. Gleason}, J. Anal. Math. 19, 171- 172 (1967; Zbl 0148.575), \textit{M. Roitman} and \textit{Y. Sternfeld}, Trans. Am. Math. Soc. 267, 111-124 (1981; Zbl 0474.46039) and the Kahane- Zelazko-paper cited above] no function theoretic arguments of any consequence are used. After that the theorem of Gleason-Kahane-Zelazko for a commutative complex Banach algebra with identity and symmetric involution (which means that every self-adjoint element has real spectrum) is deduced from the measure theorem.