Ideals with bounded approximate identities in Fourier algebras.
For a closed subgroup \(H\) of an amenable locally compact group \(G\) the ideal \(I(H)\) in the Fourier algebra \(A(G)\) of \(G\) is shown to have a bounded approximate identity, where for a subset \(E\) of \(G\) the ideal \(I(E)\) is defined by \(\{u\in A(G): u(x)= 0\) for \(x\in E\}\). Moreover, any ideal with a bounded approximate identity is of type \(I(E)\), and all these subsets \(E\) of \(G\) are characterized. Concerning bounds of approximate identities it is shown that for closed subgroups \(H\) in locally nilpotent locally compact groups or in almost connected groups \(G\) with a compact normal subgroup \(K\) such that \(G/K\) is nilpotent the ideal \(I(H)\) has an approximate identity with norm bounded by 2. Generalizations of the above results to the Herz-Figà-Talamanca algebras \(A_p(G)\) are established.
scientific article; zbMATH DE number 1992844
scientific article