If G is a locally compact
Abelian group, any subalgebra A of M(G) that contains a dense ideal of L1(G) can
be mapped homomorphically onto C(K) for any Helson set K in the dual group.
Then, by choosing a Helson set homeomorphic to the one-point compactification N∞
of the natural numbers, the ideal structure of A can be explored from known
properties of C(N∞). As special cases normed subalgebras are considered
and for them — by different techniques — information on their countably
generated, closed ideals J can be obtained. Necessarily Z(J) is open-closed;
if G is compactly generated and A contains such a nonzero J,G must be
Zn× C[ C a compact group] and J must consist of those L1 functions whose
Fourier transforms vanish on Tn× E, where E is a cofinite subset of the
dual of C. In particular, a Segal algebra on G (satisfying mild restrictions)
can have a countably generated regular maximal ideal if and only if G is
finite.
