Holomorphic Approximation to Boundary Value Algebras

have uniformly dense range! In terms of uniform algebras, it becomes_a question of necessary and sufficient conditions for thejuniform algebra H(D) of uniform limits o£ functions holomorphic near D Jo coincide with the uniform algebra A(D) of all continuous functions on D which are holomorphic on D. A necessary condition obviously is that the respective collections of complex homomorphisms, A//(Z>) and AA(D), of these algebras coincide. Since Arens [1] demonstrated for the case n = 1 that LA(D) *= D (i.e. every complex homomorphism of A(D) arises from point evaluation at points of D), and again in the case n = 1 Runge's theorem implies AH(D) = D, the necessary condition always holds on domains in the complex plane. Nevertheless, there existjelatively compact connected domains D in the complex plane for which H(D) ^ A(D). Several such examples are developed in Gamelin's treatise [17]. _Failure of approximation in the plane is_a local boundary property of Z>, since H{D) = AJ^D), if for every z G D there is a closed neighborhood Vz on which A(D n Vz) = H(D n Vg). See [17]. In 1967, Vitushkin [32] formulated a necessary and sufficient condition for approximation in terms of continuous analytic capacity; roughly speaking, approximation takes place if the boundary of D is "not too large." Following thisjone-dimensional motivation, it has frequently been conjectured that Hol(D) ^ Hol(Z)) n C(D) has dense range on any relatively compact domain of holomorphy D with "reasonable" boundary. Recent examples, which are described below, show the conjecture to be false even in the case dD is C. These counterexamples are established by showing the failure of the necessary condition-viz., AH(D)=£AA(D\ due to analytic continuation beyond^D of functions in H(D) not possible to the same extent for functions in A(D). When D is holomorphically convex-i.e. &H(D) = D,