A generalization of the Gleason-Kahane-Zelazko theorem
Among the results of this paper is Theorem 4: Let A be a commutative Banach algebra with involution \(x\to x^*\) satisfying \(\hat x{}^*=\hat x^-\). Suppose that there is an element \(x_ 0\) in A, with \(\hat x{}_ 0\) never zero, and that there is a one-to-one real-valued function \(\phi\) on the maximal ideal space \({\mathcal M}\) of A such that \(\hat x_ 0\phi^ j=\hat x_ j\) for some \(x_ j\) in A (1\(\leq j\leq n)\). If M is a subspace (not a priori closed) of codimension n in A with the property that each element in M belongs to at least n regular maximal ideals, then M is an ideal which is the intersection of n regular maximal ideals. It is then shown that this result applies to certain group algebras \(L^ 1(G)\) as well as to \(C_ 0(S)\), S a closed subset of \({\mathbb{R}}\times {\mathbb{Z}}^{\infty}\). This is a partial answer to a question of \textit{C. R. Warner} and \textit{R. Whitley} in Proc. Am. Math. Soc. 76, 263-267 (1979; Zbl 0497.46034).
commutative Banach algebra with involution
scientific article; zbMATH DE number 3851802