| In this thesis, we present some new notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual (weak* closed) algebra setting, of the basic theory of strong Morita equivalence due to Blecher, Muhly, and Paulsen. We generalize Rieffel's theory of Morita equivalence for W*-algebras to non-selfadjoint dual operator algebras. Our theory contains all examples considered up to this point in the literature of Morita-like equivalence in a dual (weak* topology) setting. Thus, for example, our notion of equivalence relation for dual operator algebras is coarser than the one defined recently by Eleftherakis.;In addition, we give a new dual 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 theory of W*-modules to the setting of non-selfadjoint algebras of Hilbert space operators which are closed in the weak* topology. That is, we find the appropriate weak* topology variant of the theory of rigged modules due to Blecher. We prove various versions of the Morita I, II, and III theorems for dual operator algebras. In particular, we prove that two dual operator algebras are weak* Morita equivalent in our sense if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak* continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak* Morita equivalence bimodule. |
