Let K and X be compact plane sets such that . Let P(K) be the uniform closure of polynomials on K, let R(K) be the uniform closure of rational functions on K with no poles in K and let A(K) be the space of continuous functions on K which are analytic on int(K). Define P(X,K),R(X,K) and A(X,K) to be the set of functions in C(X) whose restriction to K belongs to P(K),R(K) and A(K), respectively. Let S0(A) denote the set of peak points for the Banach function algebra A on X. Let S and T be compact subsets of X. We extend the Hartogs–Rosenthal theorem by showing that if the symmetric difference SΔT has planar measure zero, then R(X,S)=R(X,T) . Then we show that the following properties are equivalent:
- (i)R(X,S)=R(X,T) ;
- (ii) and ;
- (iii)R(K)=C(K) for every compact set ;
- (iv) for every open set U in ℂ ;
- (v)for every p∈X there exists an open disk Dp with centre p such that
We prove an extension of Vitushkin’s theorem by showing that the following properties are equivalent:
- (i)A(X,S)=R(X,T) ;
- (ii) for every closed disk in ℂ ;
- (iii)for every p∈X there exists an open disk Dp with centre p such that