Characterizations of peripherally multiplicative mappings between real function algebras (Q2875441)

Let \(X\) be a compact Hausdorff space. A topological involution is a continuous mapping \(\tau: X\to X\) such that \(\tau (\tau(x))=x\) for all \(x\in X\). Define \(C(X,\tau)= \{f\in C(X): f\circ \tau=\overline{f}\}\). Let \(\mathcal {A} \subset C(X,\tau)\) be a real function algebra. Given \(f\in \mathcal {A}\), the peripheral spectrum of \(f\) is the set \(\sigma_{\pi} (f)\) of spectral values of maximum modulus. The author demonstrates that if \(T_1, T_2 : \mathcal {A}\to {\mathcal{B}}\) and \(S_1, S_2 : \mathcal {A}\to \mathcal {A}\) are surjective mappings between real function algebras \(\mathcal {A} \subset C(X,\tau)\) and \({\mathcal{B}} \subset C(Y,\phi)\) that satisfy \(\sigma_{\pi} (T_1 (f) (T_2 (g))= \sigma_{\pi} (S_1 (f) (S_2 (g))\) for all \(f, g \in \mathcal {A}\), then there exists a homeomorphism \(\psi: \text{Ch}\,\mathcal{B}\to \text{Ch}\,\mathcal {A}\) between the Choquet boundaries such that \((\psi\circ \phi) (y)= (\tau\circ \psi) (y)\) for all \(y\in \text{Ch}\,\mathcal{B}\) and there exists functions \(\kappa_1, \kappa_2\in \mathcal{B}\) such that \(\kappa_1^{-1}=\kappa_2\) such that \(T_j (f)(y)= \kappa_j (y) S_j (f)(\psi(y))\) for all \(f\in \mathcal {A}\), all \(y\in \text{Ch}\,\mathcal{B}\) and \(j=1, 2\). As a consequence, it is shown that if either Ch\(\mathcal {A}\) or Ch\(\mathcal{B}\) is a minimal boundary for its corresponding algebra, then the same result holds for surjective mappings \(T_1, T_2 : \mathcal {A}\to {\mathcal{B}}\) and \(S_1, S_2 : \mathcal {A}\to \mathcal {A}\) that satisfy \(\sigma_{\pi} (T_1 (f) (T_2 (g))\cap \sigma_{\pi} (S_1 (f) (S_2 (g))\not = \emptyset\) for all \(f, g \in \mathcal {A}\).