Vol. 30, No. 1, 1969

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A characterization of regular maximal ideals

Charles Robert Warner and Robert James Whitley

Vol. 30 (1969), No. 1, 277–281

In 1967, Gleason characterized the maximal ideals in a commutative Banach algebra A with identity as those subspaces M of codimension one in A which contain no invertible elements. (Kahane and Zelazko gave the same characterization in 1968.) An equivalent statement of this property of M is:

(1) Each element of M belongs to some regular maximal ideal. The question we examine is when does this distinguish the regular maximal ideals from the other subspaces of codimension one in a commutative Banach algebra without identity.

We show that if A is generated by a single element then a closed subspace M of codimension one in A and satisfying (1) is a regular maximal ideal and we show by an example that this result may fail for an algebra which is generated two elements. We have results related to the above, which can be applied to L1(G), where G is a locally compact abelian metrizable group; a sample corollary is that a (not a priori closed) subspace M of codimension one for which (1) holds is a regular maximal ideal in L1(G).

Mathematical Subject Classification 2000
Primary: 46J20
Received: 21 November 1968
Published: 1 July 1969
Charles Robert Warner
Robert James Whitley