A generalization of a theorem on biseparating maps. (Q1873666)

Firstly, the authors prove that any order bounded linear biseparating map between two \(\Phi\)-algebras is a weighted isomorphism (Theorem 3.3). Secondly, they prove that if \(A\) and \(B\) are uniformly closed \(\Phi\)-algebras and if \[ \text{every universally \(\sigma\)-complete projection band in \(A\) is essentially one-dimensional,} \tag{1} \] then every linear biseparating map \(\varphi: A \to B\) is a weighted isomorphism (Theorem~ 5.2). Finally, the authors derive that the Riesz space \(C (X)\) has property \((1)\) provided that \(X\) is a completely regular topological space (Theorem 5.5). Combining Theorems 5.2 and 5.5, they obtain, as a corollary, the assertion that if \(X\) and \(Y\) are completely regular topological spaces, then every biseparating map \(\varphi: C (X) \to C (Y)\) is a weighted isomorphism (which is the result proved by \textit{J. Araujo}, \textit{B. Beckenstein} and \textit{L. Narici} [J. Math. Anal. Appl. 192, 258--265 (1995; Zbl 0828.47024)]). In particular, \(C (X)\) and \(C (Y)\) are isomorphic \(\Phi\)-algebras if and only if there exists a~linear biseparating map from \(C (X)\) onto \(C (Y)\).