Modules over operator algebras, and the maximal C*-dilation

We continue our study or 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. (C) 1999 Academic Press.