THE WEDDERBURN DECOMPOSABILITY OF SOME COMMUTATIVE BANACH-ALGEBRAS

In this paper we study the question whether certain non-semisimple, commutative Banach algebras have a Wedderburn decomposition U = B ⊕ rad U, where B is a subalgebra of U. Let A(G) be the Fourier algebra of a locally compact abelian group G, and let E be a closed subset of G. Let J(E) be the smallest closed ideal in A(G) whose hull is E. We prove that, when E is a set of non-synthesis, the quotient algebra A(G) J(E) never has a Wedderburn decomposition even in the purely algebraic sense. This result is extended to cover certain Beurling algebras Ax(Rk) and Ax(Tk). © 1992.