  We give a new Banach module characterization of $W^*$-modules, also known as
selfdual Hilbert $C^*$-modules over a von Neumann algebra. This leads to a
generalization of the notion, and the theory, of W*-modules, to the setting
where the operator algebras are $\sigma$-weakly closed algebras of operators on
a Hilbert space. That is, we find the appropriate weak* topology variant of our
earlier notion of {\em rigged modules}, and their theory, which in turn
generalizes the notions of C*-module, and Hilbert space, successively. Our {\em
w*-rigged modules} have canonical `envelopes' which are W*-modules. Indeed,
w*-rigged modules may be defined to be a subspace of a W*-module possessing
certain properties.
