On the maximal ideal space of Dales–Davie algebras of infinitely differentiable functions

Let X be a perfect, compact plane set, and let M =( M n ) be a sequence of positive numbers such that M 0 =1 andM n / M k M n − k ⩾ ( )for all n ∈ℕ and k =0, 1, 2, …, n . We consider a remarkable class of Banach function algebras of infinitely differentiable functions f on X such that ∑ n = 0 ∞ | | f ( n ) | | X / M n < ∞ These algebras, called Dales–Davie algebras, are denoted by D ( X , M ), and they are complete under certain conditions on X . The main aim of this work is to find conditions on the sequence M =( M n ) to guarantee that D ( X , M ) is natural; that is, its maximal ideal space is identified with X . We present a general result on the naturality of D ( X , M ) using some formulas from combinatorial analysis. In particular, it is shown that if X is uniformly regular, and if the sequence ( P n )=( M n / n !) satisfies any one of the following conditions, then D ( X , M ) is natural: (i) sup { P i P j / P i + j −1 : i , j ∈ℕ}<∞; (ii)P n 2 ⩽ P n − 1P n + 1for all n ∈ℕ; and (iii) B n =max { P k P n − k / P n : 1⩽ k ⩽ n −1}→0 as n →∞.