  We consider a variant of the notion of Morita equivalence appropriate to
weak* closed algebras of Hilbert space operators, which we call {\em weak
Morita equivalence}. We obtain new variants, appropriate to the dual algebra
setting, of the basic theory of strong Morita equivalence, and new
nonselfadjoint variants of aspects of Rieffel's $W^*$-algebraic Morita
equivalence.
