IDEMPOTENTS IN QUOTIENTS AND RESTRICTIONS OF BANACH-ALGEBRAS OF FUNCTIONS

Let A(beta) be the Beurling algebra with weight (1+\n\)(beta) on the u nit circle T and, for a closed set E subset of or equal to T, let J(A beta) (E) = {f epsilon A(beta) : f = 0 on a neighbourhood of E}. 1 We prove that, for beta > 1/2, there exists a closed set E subset of or e qual to T of measure zero such that the quotient algebra A(beta)/<(J(A beta)(E))over bar> is not generated by its idempotents, thus contrast ing a result of Zouakia. Furthermore, for the Lipschitz algebras lambd a gamma and the algebra AC of absolutely continuous functions on T, we characterize the closed sets E subset of or equal to T for which the restriction algebras lambda gamma(E) and AC(E) are generated by their idempotents.