Certain Subalgebras of Lipschitz Algebras of In nitely Di erentiable Functions and Their Maximal Ideal Spaces

We study an interesting class of Banach function algebras of in nitely di erentiable functions on perfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, called Lipschitz algebras of in nitely di erentiable functions and denoted by Lip(X;M;), where X is a perfect, compact plane set, M = fMng1n =0 is a sequence of positive numbers such that M0 = 1 and (m+n)! Mm+n  (m! Mm)(n! Mn) for m; n 2 N [f0g and 2 (0; 1]. Let d = lim sup(n! Mn) 1 n and Xd = fz 2 C: dist(z;X)  dg. Let LipP;d(X;M;)[LipR;d(X;M;)] be the subalgebra of all f 2 Lip(X;M;) that can be approximated by the restriction to Xd of polynomials [rational functions with poles o Xd]. We show that the maximal ideal space of LipP;d(X;M;) is cXd, the polynomially convex hull of Xd, and the maximal ideal space of LipR;d(X;M;) is Xd, for certain compact plane sets.. Using some formulae from combinatorial analysis, we nd the maximal ideal space of certain subalgebras of Lipschitz algebras of in nitely di erentiable functions.