AMENABILITY AND SEMISIMPLICITY FOR 2ND DUALS OF QUOTIENTS OF THE FOURIER ALGEBRA A(G)

Let F subset of G be closed and A(F) = A(G)/I-F. If F is a Helson set then A(F)* is an amenable (semisimple) Banach algebra. Our main resul t implies the following theorem: Let G be a locally compact group, F s ubset of G closed, a is an element of G. Assume either (a) For some no n-discrete closed subgroup H, the interior of F boolean AND aH in aH i s non-empty, or (b) R subset of G, S subset of R is a symmetric set an d aS subset of F. Then A(F)* is a non-amenable non-semisimple Banach algebra. This raises the question: How 'thin' can F be for A(F)* to r emain a non-amenable Banach algebra?