An analogoue of Hoffman-Wermer theorem for a real function algebra (Q1108552)

Let X be a compact Hausdorff space and C(X) the Banach algebra of all complex-valued continuous functions on X. Let \(\tau\) :X\(\to X\) be a homeomorphism such that \(\tau^ 2\) is the identity on X and let \(C(X,\tau)=\{f\in C(X):f(\tau (x))=\bar f(x)\) for each \(x\in X\}\) and A a real function algebra on (X,\(\tau)\), that is, A is a uniformly closed subalgebra of C(X,\(\tau)\) contaning real constants and separating points of X. Under this setting, the authors studied an analogue of the Hoffman- Wermer theorem. That is, they proved that \(A=C(X,\tau)\) if Re A\(=\{Re f:f\in A\}\) is uniformly closed, making use of an analogoue of Bishop's theorem obtained earlier by them. Further, applying this result, they obtained the following: Let A be a uniformly closed real subalgebra of C(X) containing real constants and separating points of X. If Re A is uniformly closed in C(X), then there exists a closed subset Z of X such that \(A=\{f\in C(X):f| Z\) is real\(\}\), which can be regarded as a stronger version of the classical Hoffman-Wermer theorem.