On the Ideal Structure of Banach Algebras

For Banach algebras A in a class which includes all group and function algebras, we show that the family of ideals of A with the same hull is typically quite large, containing ascending and descending chains of arbitrary length through any ideal in the family, and that typically a closed ideal of A whose hull meets the Silov boundary of A cannot be countably generated algebraically. Guided by the results of [6], we explore here some consequences of the Cohen factorization theorem [3] for the ideal structure of a commutative Banach algebra A. Ii we denote the maximal ideal space of A by MÍ, and define the zero set of an ideal / of A to be Z(/) =D ,Z(/) = / _1(0): f £l\ C%A, we develop results which indicate that typically the class of ¡deals of A with the same zero set is quite large, containing ascending and descending chains of arbitrary length, and that typically a closed ideal of A whose zero set meets the Silov boundary of A cannot be (algebraically) countably generated. For example, if G is a locally compact Abelian group, there is an ideal of L (G) strictly between any two ideals / C¿ J which have the same zero set whenever either / or / is closed (compare [17, 7.7.2]). Further, a closed ideal j oí L (G) can be countably generated only if Zij) is open-closed, and a maximal ideal can be countably generated only if G is finite (compare [11, 2.1], [5, Corollary 3])Algebraic applications of Cohen's theorem have been few to date (principally [4, 4.7]), but because the result transforms a purely analytic condition on A (existence of a bounded approximate identity) into a purely algebraic conclusion (factorization of elements), it provides a most appropriate tool for just such applications. 1. Chains of ideals. Let A be a commutative algebra(') over a field F. An ideal of A will be a subspace over F closed under multiplication from A.HI and / are ideals, // will denote 1/g: / £ I, g £ ] !, not the ideal generated by this set. We begin with a purely algebraic remark. Presented to the Society, December 4, 1970; received by the editors December 10, 1970 and, in revised form, September 30, 1971. AMS 1970 subject classifications. Primary 46J20, 22D15, 13E99; Secondary 46J10, 43A45.