On the maximal ideal space of extended polynomial and rational uniform algebras

Authors

1 Department of Mathematics, Faculty of Science, Arak University, Arak, 38156- 8-8349, Iran.

2 Faculty of Mathematical Sciences and Computer Engineering, Teacher Train- ing University, 599 Taleghani Avenue, Tehran, 15618, I.R. Iran.

Abstract

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. De ne 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 con-
sidering 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.

Keywords