Mappings onto multiplicative subsets of function algebras and spectral properties of their products (Q496814)

Let \(A\) and \(B\) be function algebras on locally compact Hausdorff spaces \(X\) and \(Y\), \({\mathcal J}_i\) be arbitrary sets. Let \(S_i: {\mathcal J}_i \to {\mathcal S}_i\), \(T_i: {\mathcal J}_i \to {\mathcal T}_i\), \(i=1, 2\), be surjective (not necessarily linear) maps onto multiplicative subsets \({\mathcal S}_i \subset A\), \({\mathcal T}_i \subset B\). In this paper, the authors characterise mappings \(S_i\) and \(T_i\) under the conditions on the peripheral spectra of their products \[ \sigma_{\pi}(S_1(a)S_2(b))\subset \sigma_{\pi}(T_1(a)T_2(b)) \quad \text{and} \quad \sigma_{\pi}(S_1(a)S_2(b))\cap \sigma_{\pi}(T_1(a)T_2(b))\neq \emptyset \] for \(a\in {\mathcal J}_1\) and \(b\in {\mathcal J}_2\). Mainly, they show that the mappings in the first pair equal the mappings in the second one up to certain weighted composition operators on the corresponding Choquet boundaries. As a direct consequence, the authors obtain a large amount of previous results about mappings subject to various spectral conditions.