Vol. 31, No. 2, 1969

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Semi-square-summable Fourier-Stieltjes transforms

Irving Leonard Glicksberg

Vol. 31 (1969), No. 2, 367–372

For G a locally compact abelian group with dual Γ, let μ be a (finite regular Borel) measure on G with Fourier-Stieltjes transform μ. Doss has recently shown that when Γ is (algebraically) a totally ordered abelian group and μ is square integrable on the negative half Γ of Γ then its singular component σ has σ = 0 on Γ; in particular μE = 0 for each common null set E of the analytic measures (those with transforms 0 on Γ), such E being Haar-null.

In the similar (but usually distinct) case in which Γ is partially ordered by a nonzero homomorphism ψ : Γ R with Γ = ψ1(− ∞, 0] the common null sets E are known, and our purpose is to note in this setting how function algebra results apply to show μE = 0 when μ L2), and when μ salisfies sometimes weaker (but more obscure) hypotheses.

Mathematical Subject Classification
Primary: 42.52
Received: 23 December 1968
Published: 1 November 1969
Irving Leonard Glicksberg