Distance formulas in group algebras

Abstract

Let G be a locally compact amenable group, A(G) and B(G) be the Fourier and the Fourier–Stieltjes algebra of G, respectively. For a given u∈B(G), let Eu:={g∈G:|u(g)|=1}. The main result of this paper particularly states that if ‖u‖B(G)≤1 and u(Eu)‾ is countable (in particular, if Eu is compact and scattered), thenlimn→∞⁡‖unv‖A(G)=dist(v,IEu), ∀v∈A(G), where IEu={v∈A(G):v(g)=0, ∀g∈Eu}.