On the Maximal Ideal Space of the Extended Polynomial and Rational Uniform Algebras

Let K and X be compact plane sets such that K X. Let P(K) be the uniform closure of polynomials on K. Let R(K) be the closure of rational functions K with poles o K. Dene P(X;K) and R(X;K) to be the uniform algebras of functions in C(X) whose restriction to K belongs to P(K) and R(K), respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) such that fjK = 0. In this paper, we show that every nonzero complex homomorphism ' on CZ(X;K) is an evaluation homomorphism ez for some z in XnK. By considering this fact, we characterize the maximal ideal space of the uniform algebra P(X;K). Moreover, we show that the uniform algebra R(X;K) is natural.