Modules over Operator Algebras, and the Maximal C*-Dilation

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.