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).
