Failure of rational approximation on some Cantor type sets

Abstract

Let A(K) be the algebra of continuous functions on a compact set K C C which are analytic on the interior of K, and let R(K) be the closure (with respect to uniform convergence on K) of the functions that are analytic on a neighborhood of K. A counterexample of a question posed by A. O'Farrell about the equality of the algebras R(K) and A(K) when K = (K 1 x [0,1]) U ([0,1] x K 2 ) C C, with K 1 and K 2 compact subsets of [0,1], is given. Also, the equality is proved with the assumption that K 1 has no interior.