Let A bs a weak-*Dirichlet
algebra of L∞(m) and let Hx(m) denote the weak-*closure of A in L∞(m). Muhly
showed that if H∞(m) is an integral domain, then H∞(m) is a maximal weak-*closed
subalgebra of L∞(m). We show in this paper that if H∞(m) is not maximal as a
weak-*closed subalgebra of L∞(m), there is no algebra which contains H∞(m) an is
maximal among the proper weak-*closed subalgebras of L∞(m). Moreover, we
investigate the weak-*closed superalgebras of A and we try to classify them. We show
that there are two canonical weak-*closed superalgebras of A which play an
important role in the problem of describing all the weak-*closed superalgebras of
A.
