%0 Journal Article
%T Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces
%J International Journal of Nonlinear Analysis and Applications
%I Semnan University
%Z 2008-6822
%A Alimohammadi, D.
%A Nezamabadi, F.
%D 2014
%\ 01/01/2014
%V 5
%N 1 (Special Issue)
%P 9-22
%! Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces
%K Infinitely differentiable functions
%K Function algebra
%K Lipschitz algebra
%K Maximal ideal space
%K Star-shaped set
%K Uniformly regular
%R 10.22075/ijnaa.2014.111
%X We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, called Lipschitz algebras of infinitely differentiable functions and denoted by $Lip(X,M, \alpha)$, where $X$ is a perfect, compact plane set, $M =\{M_n\}_{n=0}^\infty$ is a sequence of positive numbers such that $M_0 = 1$ and $\frac{(m+n)!}{M_{m+n}}\leq(\frac{m!}{M_m})(\frac{n!}{M_n})$, for $m, n \in\mathbb{N} \cup\{0\}$ and $\alpha\in (0, 1]$. Let $d =\lim \sup(\frac{n!}{M_n})^{\frac{1}{n}}$ and $X_d =\{z \in\mathbb{C} : dist(z,X)\leq d\}$. Let $Lip_{P,d}(X,M, \alpha)$ [$Lip_{R,d}(X,M \alpha)$] be the subalgebra of all $f \in Lip(X,M,\alpha)$ that can be approximated by the restriction to $X_d$ of polynomials [rational functions with poles $X_d$]. We show that the maximal ideal space of $Lip_{P,d}(X,M, \alpha)$ is $\widehat{X_d}$, the polynomially convex hull of $X_d$, and the maximal ideal space of $Lip_{R,d}(X,M \alpha)$ is $X_d$, for certain compact plane sets. Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Lipschitz algebras of infinitely differentiable functions.
%U https://ijnaa.semnan.ac.ir/article_111_3aee2736a32d307e34b4d8bc34fafb5a.pdf