A characterization of the closed unital ideals of the Fourier-Stieltjes algebra \(B\)(\(G\)) of a locally compact amenable group \(G\).
Let \(G\) be a locally compact abelian group, \(L^1(G)\) be its group algebra and \(M(G)\) be its measure algebra. Let \(\mu\in M(G)\) be a given measure. The Glicksberg-Host-Parreau theorem [cf. \textit{B. Host} and \textit{F. Parreau}, Ann. Inst. Fourier 28, 143--164 (1978; Zbl 0368.43001)] states that the ideal \(\mu\ast L^1(G)\) is closed in \(L^1(G)\) iff \(\mu\) is the product of an invertible measure and an idempotent measure. This theorem is equivalent to the statement: A closed ideal \(I\) of \(M(G)\) is unital iff it is of the form \(I=\mu\ast M(G)\) for some measure \(\mu\in M(G)\). The aim of this paper is to establish a similar theorem for the Fourier-Stieltjes algebra \(B(G)\) of a locally compact amenable group \(G\). The author presents the following result: A closed ideal \(I\) of \(M(G)\) is unital iff \(I\) is a principal ideal. -- The paper also contains an abstract version of this result.
scientific article; zbMATH DE number 2037124
scientific article