Document Type : Research Paper

Authors

• S. Moradi ¹
• T. G. Honary ²
• D. Alimohammadi ¹

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 $K\subseteq X$. 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' on $CZ(X,K)$ is an evaluation homomorphism $e_z$ for some $z$  in $X\setminus K$. 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.

Keywords

• Maximal ideal space
• uniform algebras
• nonzero complex homomorphism