Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 3 2 2012 06 01 On the maximal ideal space of extended polynomial and rational uniform algebras 1 12 32 10.22075/ijnaa.2012.32 EN S. Moradi Department of Mathematics, Faculty of Science, Arak University, Arak, 38156- 8-8349, Iran. T. G. Honary Faculty of Mathematical Sciences and Computer Engineering, Teacher Train- ing University, 599 Taleghani Avenue, Tehran, 15618, I.R. Iran. D. Alimohammadi Department of Mathematics, Faculty of Science, Arak University, Arak, 38156- 8-8349, Iran. Journal Article 2011 06 13 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. https://ijnaa.semnan.ac.ir/article_32_ded7ad00ddc06fb990aa09ff3ab151bd.pdf