  We continue our study of the general theory of possibly nonselfadjoint
algebras of operators on a Hilbert space, and modules over such algebras,
developing a little more technology to connect `nonselfadjoint operator
algebra' with the C$^*-$algebraic framework. More particularly, we make use of
the universal, or maximal, C$^*-$algebra generated by an operator algebra, and
C$^*-$dilations. This technology is quite general, however it was developed to
solve some problems arising in the theory of Morita equivalence of operator
algebras, and as a result most of the applications given here (and in a
companion paper) are to that subject. Other applications given here are to
extension problems for module maps, and characterizations of C$^*-$algebras.
