Vol. 51, No. 1, 1974

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Ideals in convolution algebras on Abelian groups

William Erb Dietrich

Vol. 51 (1974), No. 1, 75–88

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.

Mathematical Subject Classification 2000
Primary: 43A10
Received: 4 October 1972
Revised: 28 February 1973
Published: 1 March 1974
William Erb Dietrich