In this paper, we describe the powers of maximal ideals in the measure algebra of some locally compact Abelian groups in terms of the derivatives of the Fourier–Laplace transform of compactly supported measures. We show that if the locally compact Abelian group has sufficiently many real characters, then all derivatives of the Fourier–Laplace transform of a measure at some point of its spectrum completely characterize the measure. We also show that the derivatives of the Fourier–Laplace transform of a measure can be used to describe the powers of the maximal ideals corresponding to the points of the spectrum of the measure on discrete Abelian groups with finite torsion-free rank.
László Székelyhidi. "On the powers of maximal ideals in the measure algebra." Banach J. Math. Anal. 10 (2) 385 - 399, April 2016. https://doi.org/10.1215/17358787-3495693