Wermer's well-known maximality theorem [4], [5] has been generalized by Hoffman and Singer [2] as follows. Let G be an ordered abelian group with nonnegative class G+, and dual group P. Then the subalgebra A of C(F) generated by G+ can be extended to a maximal subalgebra of C(F), provided there exists a homomorphism t:#90 of G into the additive group R with t < 0 on G+. The homomorphism t is unique to within multiplication by a positive scalar, and the maximal subalgebras so obtained are translates of each other, as set forth precisely in [2]. (See [1, pp. 193-194] for an interesting investiga tion of maximality. A is a Dirichlet algebra, [1].) Our theorem is a converse: suppose that B is a maximal proper closed subalgebra of C(F), or of li(G), and that for each element g of G, either gEB or g-'eB. Of course, l1(G) is construed as a (dense) subalgebra of C(F) via the Fourier transform.