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

We continue our study or the general theory of possibly nonselfadjoint alge bras of operators on a Hilbert space, and modules. over such algebras, deve loping a little more technology to connect "nonselfadjoint operator algebra " with the C*-algebraic framework. More particularly, we make use of the un iversal, or maximal, C*-algebra generated by an operator algebra, and C*-di lations. This technology is quite general, however it was developed to solv e some problems arising. in the theory of Morita equivalence of operator al gebras, and as a result most of the applications given here (and in a compa nion paper) are to that subject. Other applications given here are to exten sion problems for module maps. and characterizations of C*-algebras. (C) 19 99 Academic Press.