Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces

Alimohammadi, D.; Nezamabadi, F.

doi:10.22075/ijnaa.2014.111

Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces

Document Type : Research Paper

Authors

D. Alimohammadi

F. Nezamabadi

Department of Mathematics, Faculty of Science, Arak University, P. O. Box: 38156-8-8349, Arak, Iran.

10.22075/ijnaa.2014.111

Abstract

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.

Keywords

Infinitely differentiable functions

Function algebra

Lipschitz algebra

Maximal ideal space

Star-shaped set

Uniformly regular