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 off k. define 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 f|k = 0. in this paper, we show that every nonzero complex homomorphism φ on cz(x,k) is an evaluation homomorphism ez for some z in xk. 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.