Norm-linear and norm-additive operators between uniform algebras
Let \(A\subseteq C(X)\) and \(B\subseteq C(Y)\) be uniform algebras with Choquet boundaries \(\delta A\) and \(\delta B\), respectively, and let \({\mathbb R}_+\) denote the set of nonnegative reals numbers. The authors call a map \(T:A\to B\) norm-additive in modulus if \[ \big\| |Tf|+|Tg|\big\|=\big\| |f|+|g|\big\|\qquad (f,g\in A). \] It is then shown that an \({\mathbb R}_+\)-homogeneous, norm-additive in modulus surjection \(T:A\to B\) is induced on the Choquet boundary by a continuous map \(\tau:\delta A\to\delta B\) in the sense that \(|(Tf)(\tau(x))|=|f(x)|\) for every \(x\in \delta A\) and every \(f\in A\). Actually, \(\tau\) is a homeomorphism whenever \(T\) is bijective. The key idea is to show that, for each \(x\in\delta A\), the set \[ E_x:=\bigcap_{f\in A\atop |f(x)|=\|f\|} |T(f)|^{-1}\{\|Tf\|\} \] is a singleton and belongs to \(\delta B\). Several applications follow where norm conditions on a surjection \(T:A\to B\), which force it to be an algebra isomorphism, are studied. The list is too long to be given here, but as a sample example, we note that it is shown that if a surjection \(T\) between uniform algebras satisfies (i) \(\|Tf+\alpha Tg\|=\|f+\alpha g\|\), \(|\alpha|=1\), and (ii) \(T\) is unital and \(T(\sqrt{-1})=\sqrt{-1}\), then \(T\) is an isometric algebra isomorphism. The same conclusion holds if (ii) is replaced by (ii') \(T\) preserves the peripheral spectrum of \(\mathbb C\)-peaking functions from \(A\).
scientific article; zbMATH DE number 5568968