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

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 ۱۹۹۹, 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=۰}^\infty is a sequence of positive numbers such that M_۰ = ۱ and \frac{(m+n)!}{M_{m+n}}\leq(\frac{m!}{M_m})(\frac{n!}{M_n}), for m, n \in\mathbb{N} \cup\{۰\} and \alpha\in (۰, ۱]. Let d =\lim \sup(\frac{n!}{M_n})^{\frac{۱}{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