A characterization of C(X) among algebras on planar sets by the existence of a finite universal Korovkin system (Q1086451)

A subset T of a uniform algebra A is called a universal Korovkin system for A, iff for every uniform algebra B, every unital algebra homomorphism \(L: A\to B\) and every net \(L_{\alpha}: A\to B\) of linear contractions, the convergence \(L_{\alpha}g\to Lg\) (g\(\in T)\) already implies the convergence \(L_{\alpha}f\to Lf\) for every \(f\in A.\) Let X be a compact subset of \({\mathbb{C}}\). Then a well-known version of the classical Korovkin theorem yields the existence of a finite universal Korovkin system T for the algebra C(X) of all continuous \({\mathbb{C}}\)-valued functions on X, namely \(T=\{1,z,| z| ^ 2\}\). The purpose of the paper is to show that this property in fact characterizes C(X) among the following uniform algebras associated with \(X: P(X)=\{f\in C(X):\) f can be approximated uniformly on X by polynomials\(\}\) ; \(R(X)=\{f\in C(X):\) f can be approximated uniformly on X by rational functions with poles off \(X\}\) ; \(A(X)=\{f\in C(X):\) f is holomorphic on the interior of \(X\}\). Suppose that A is one of these uniform algebras. Then it is shown that A possesses a finite universal Korovkin system iff \(A=C(X)\). As an application, the algebra C(X) (X\(\subset {\mathbb{C}}\) compact, polynomially convex and nowhere dense) is characterized as the only singly generated uniform algebra possessing a finite universal Korovkin system.