Strong and Extremely Strong Ditkin sets for
the Banach Algebras Apr(G) = Ap ⋂ Lr(G)
Abstract Let Ap(G) be the Figa-Talamanca, Herz Banach Algebra on G; thus A2(G) is the Fourier algebra. Strong Ditkin (SD) and Extremely Strong Ditkin (ESD) sets for the Banach algebras Apr (G) are investigated for abelian and nonabelian locally compact groups G. It is shown that SD and ESD sets for Ap(G) remain SD and ESD sets for Apr(G), with strict inclusion for ESD sets. The case for the strict inclusion of SD sets is left open.A result on the weak sequential completeness of A2(F) for ESD sets F is proved and used to show that Varopoulos, Helson, and Sidon sets are not ESD sets for A2r(G), yet they are such for A2(G) for discrete groups G, for any 1 ≤ r ≤ 2.A result is given on the equivalence of the sequential and the net definitions of SD or ESD sets for σ-compact groups.The above results are new even if G is abelian.