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Bulletin of the London Mathematical Society Advance Access originally published online on December 11, 2007
Bulletin of the London Mathematical Society 2007 39(6):940-948; doi:10.1112/blms/bdm084
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© 2007 London Mathematical Society

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

M. Abtahi

Department of Mathematics
Damghan University of Basic Sciences
Damghan
I.R. IRAN
abtahi@dubs.ac.ir

T. G. Honary

Faculty of Mathematical Sciences and Computer Engineering
Teacher Training University
599 Taleghani Avenue
Tehran 15618
I.R. IRAN

Received 30 September 2005. Revision received 24 May 2007.

Let X be a perfect, compact plane set, and let M=(Mn) be a sequence of positive numbers such that M0=1 and Mn/MkMn–k≥ Formula for all nisinN and k=0, 1, 2, ..., n. We consider a remarkable class of Banach function algebras of infinitely differentiable functions f on X such that Formula 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=(Mn) 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 (Pn)=(Mn/n!) satisfies any one of the following conditions, then D(X, M) is natural: (i) sup {PiPj/Pi+j–1:i, jisinN}<{infty}; (ii) Formula ; and (iii) Bn=max {PkPn–k/Pn: 1≤k≤n–1}->0 as n->{infty}.

2000 Mathematics Subject Classification 46J10, 46J15, 46J20.


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