TY - JOUR
ID - 111
TI - Certain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces
JO - International Journal of Nonlinear Analysis and Applications
JA - IJNAA
LA - en
SN -
AU - Alimohammadi, D.
AU - Nezamabadi, F.
AD - Department of Mathematics, Faculty of Science, Arak University, P. O. Box: 38156-8-8349, Arak, Iran.
Y1 - 2014
PY - 2014
VL - 5
IS - 1 (Special Issue)
SP - 9
EP - 22
KW - Infinitely differentiable functions
KW - Function algebra
KW - Lipschitz algebra
KW - Maximal ideal space
KW - Star-shaped set
KW - Uniformly regular
DO - 10.22075/ijnaa.2014.111
N2 - 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.
UR - https://ijnaa.semnan.ac.ir/article_111.html
L1 - https://ijnaa.semnan.ac.ir/article_111_3aee2736a32d307e34b4d8bc34fafb5a.pdf
ER -