A note on the peak points for real function algebras (Q914126)

Let X be a compact Hausdorff space and \(\tau\) an involution on X (i.e. \(\tau\) is a homeomorphism on X such that \(\tau^ 2\) is the identity map on X). Let C(X) denote the complex Banach algebra of all continuous complex-valued functions on X and put \[ C(X,\tau)=\{f\in C(X):\;f(\tau (x))=\overline{f(x)}\text{ for all } x\in X\}. \] A real function algebra on X is a uniformly closed real subalgebra of C(X,\(\tau\)) which contains the constant functions and separates the points of X. The author considers the set of peak points and the set of pairs of peak points for a real function algebra A and shows that in the metrizable case the union of these two sets is a boundary for A and a \(G_{\delta}\)-set.