DAY POINTS FOR QUOTIENTS OF THE FOURIER ALGEBRA A(G), EXTREME NONERGODICITY OF THEIR DUALS AND EXTREME NON ARENS REGULARITY

摘要

Let J be a closed ideal of the Fourier algebra A = A(G) of the metrisable locally compact group G, with identiy e, and F = Z(J) is contained in G its zero set. G need not be abelian, yet the results that follow are new even if G = R or T (the real line or the torus). Let PM(G) = A(G). Call a ∈ F aMahlonM. Day point of J and let D_1 (J) be the set of all such, if thereis a sequence u_n ∈ A ∩ C_c(G) such that (ⅰ) 1 = u_n(a) = ‖u_n‖,(ⅱ) for any neighborhood V of a thereis some k such that F ∩ supp u_n is contained in F ∩ V if n ≥ k and (ⅲ) {u_n} is a Sidon sequence in A/J, i.e. there is some d > 0 such that ‖ Σ_1~n α_j u_j ‖A/J ≥ d Σ_1~n |α_j| for all complex α_j and n ≥ 1.