Result: Pervasive Algebras of Analytic Functions: Pervasive algebras of analytic functions

Let \(X\) be a compact Hausdorff space and \(S\) a complex or real closed subspace of \(C(X,\mathbb{C})\) or \(C(X,\mathbb{R})\) respectively, and let \(Y\) be a closed subset of \(X\). \(S\) is said to be complex or real pervasive on \(Y\) if the functions of \(S\) restricted to \(E\) are dense in \(C(E,\mathbb{C})\) or \(C(E,\mathbb{R})\), respectively for each proper closed subset \(E\) of \(Y\). These properties are investigated for the case where \(X\) is an open proper subset \(U\) of the Riemann sphere \(\widehat{\mathbb{C}}\) and \(S\) is the algebra \(A(U)\) of all complex valued functions continuous on \(\widehat{\mathbb{C}}\) and analytic on \(U\), or \(S= \text{Re }A(U)\), respectively, and it is supposed that \(U\) has no inessential boundary points -- i.e. points where all functions of \(A(U)\) can be extended analytically. Then \(A(U)\) is complex pervasive on \(\partial U\) if and only if \(\partial U_i=\partial U\) for each component \(U_i\) of \(U\). If \(\text{Re }A(U)\) is real pervasive on \(\partial U\) then \(U\) has at most one component \(U_k\) that is not simply connected, and in this case \(\partial U_k=\partial U\). If on the other hand \(U\) has at least one component \(U_k\) such that \(\partial U_k=\partial U\) then \(\text{Re }A(U)\) is real pervasive on \(\partial U\). In case that all components \(U_i\) of \(U\) are simply connected and \(\partial U_i\neq\partial U\), then the real pervasiveness is characterized in terms of properties of the boundary points.