Some Convergence Theorems in Fourier Algebras

Abstract

Let G be a locally compact amenable group and A(G) and B(G) be the Fourier and the Fourier-Stieltjes algebras of G; respectively. For a power bounded element u of B(G), let epsilon(u) : = {g is an element of G : |u(g)| = 1}. We prove some convergence theorems for iterates of multipliers in Fourier algebras. (a) If parallel to u parallel to(B(G)) <= 1, then lim(n ->infinity) parallel to u(n)v parallel to(A(G)) = dist(v, I epsilon(u)) for v is an element of A(G), where I-epsilon u = {v is an element of A(G) : v(epsilon(u)) = {0}}. (b) The sequence {u(n)v}(n is an element of N) converges for every v is an element of A(G) if and only if epsilon(u) is clopen and u(epsilon(u)) = {1}. (c) If the sequence {u(n)v}(n is an element of N) converges weakly in A(G) for some v is an element of A(G), then it converges strongly.