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Several characterizations are given for the Banach algebra of (n 1)-times continuously differentiable functions whose (n 1)st derivative satisfles a bounded Lipschitz condition. The structure of the closed primary ideals is investigated and spectral synthesis is shown to be satisfied. 0. Introduction. The aim of this paper is to study the ideal structure of the subalgebra of Dn-1 [0, 1] (the algebra of (n 1)-times continuously differentiable functions on [0, 1]) consisting of functions whose (n l)st derivative satisfies a bounded Lipschitz condition. We present several characterizations of this algebra and then show that every closed ideal I is the intersection of primary ideals determined by I; i.e., that spectral synthesis holds. We show further that the closed primary ideals are determined by the differentiability properties of the algebra. We let 0" (the interval [0, 1] is fixed throughout) be the algebra of functions f for which there exists a sequence {fp}p1= of functionsp E Dn such that IIfE, -fIIl..1 -f 0 and suppIIfpIIn <oo where 1 Ilr = I=0rlf(')jjo/i! is the standard Banach algebra norm on Dr. Vn is a commutative Banach algebra under the norm llfll = inf {suPIpIn: fp EDn and IIfJ -fII_1 ? 0) Dn is easily seen to be regular since we have Dn c Dn C Dn-1 by Theorem 1.3. In [7] D. Sherbert studies Banach algebras of Lipschitz functions on metric spaces. His work includes V' as a special case although the theorem on spectral synthesis was obtained later by Waelbroeck [8] and independently by G. Glaeser [3]. Henceforth we refer to Dn as D. 1. Functions with bounded nth-order differences. This section is devoted to the proof of Theorem 1.3 which characterizes V as an algebra of functions Received by the editors June 21, 1975. AMS (MOS) subject classifications (1970). Primary 46J15, 46J20; Secondary 39A05.

A Banach algebra of functions with bounded 𝑛th differences