Weighted composition operators and the Gleason-Kahane-Zelazko theorem
The author shows that if \(A\) and \(B\) are uniform algebras, respectively on compact Hausdorff spaces \(M\) and \(N\), then any linear isometry of \(A\) onto \(B\) is represented by a weighted composition operator involving a homeomorphism between the Gelfand spectra of \(A\) and \(B\). Further, he considers the case when \(A\) and \(B\) are the (\(m\)-convex) algebras \(H(\mathbb{C}^n)\) and \(H(\mathbb{C}^m)\) of entire functions endowed with the compact open topology. In this case, any continuous linear operator from \(A\) into \(B\), mapping the invertible elements of \(A\) into invertible elements of \(B\), is also represented as a composition operator defined by means of a holomorphic map from \(\mathbb{C}^n\) into \(\mathbb{C}^n\). A related result is obtained in the case when \(A = C(M)\), \(B = C(N)\), where \(M\) and \(N\) are locally compact and countably compact, and \(A\), \(B\) are endowed with the compact open topology. The motivation of this paper is the investigation of the Banach-Stone and Holsztyński theorems in the context of locally multiplicatively convex algebras. The proofs are based upon a generalization of the Gleason-Kahane-Żelazko theorem to sequentially complete \(m\)-convex algebras, also obtained in the paper under review.
scientific article; zbMATH DE number 2128867