Ideals of Finite Codimension in C[ 0, 1 ] and L 1 (R)

Abstract

Let M be a subspace of codimension 1 in a commutative Banach algebra with identity. It was shown in [6] and [8] that if each element x in M belongs to a maximal ideal I., which may depend on x, then M is itself a maximal ideal. This interesting result as first proved depended on the Hadamard Factorization Theorem: later proofs used a one-sided Liouville Theorem [10], [1, p. 51], [2, Problem 11, p. 111], [4, Lemma 32, p. 1043]. This characterization of maximal ideals was extended in [11] to algebras without identity. The main results were: (1) [11, Theorem 2]: Let A be a commutative Banach algebra with one generator. If a closed subspace M of codimension 1 in A has the property that each element in M belongs to a regular maximal ideal, then M is a regular maximal ideal. (The proof given in [11] is in error, but it can be corrected.) An example is given of an algebra with two generators in which this characterization does not hold. (2) [11, Theorems 4 and 5]. Let G be a locally compact Abelian group with G sigma-compact. Then a subspace M of codimension 1 in L'(G) which has the property that each element in M belongs to some regular maximal ideal must itself be a regular maximal ideal. (If G is not sigma-compact, then each Fourier transform vanishes and so each subspace of codimension 1 has the property which is meant to characterize maximal ideals.) This result is also a corollary of the more basic result below.