A Morita Theorem for Dual Operator Algebras

We prove that two dual operator algebras are weak Morita equivalent in the sense of [4] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weakcontinuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case we can characterize such functors as the module normal Haagerup tensor product with an appropriate weak Morita equivalence bimodule. We also show that the category of dual operator modules over a W -algebra generated by a dual operator algebra M , is a subcategory of the category of dual operator modules over M . We develop the theory of the W -dilation which connects the non-selfadjoint dual operator algebra with the W -algebraic framework. In particular we use the maximal W -algebra generated by a dual operator algebra and show that every dual operator module is a weak-closed submodule of its ‘maximal dilation’. Indeed, in the case of weak Morita equivalence this maximal dilation turns out to be a W -module. The theory of the W -dilation is a key part of the proof of our main theorem. 1. notation and Introduction By a dual operator algebra, we will mean a unital weak-closed algebra of operators on a Hilbert space which is not necessarily selfadjoint. One can view a dual operator algebra as a non-selfadjoint analogue of a von Neumann algebra. By a nonselfadjoint version of Sakai’s theorem (see e.g. Section 2.7 in [7]), a dual operator algebra is characterized as a unital operator algebra which is also a dual operator space. In [4], the author and Blecher introduced some new variants of the notion of Morita equivalence appropriate to dual operator algebras. We proved therein that two dual operator algebras which are weak Morita equivalent in our sense, have equivalent categories of dual operator modules. In the present work, we prove the converse: if two dual operator algebras have equivalent categories of dual operator modules then they are weak-Morita equivalent in the sense of [4]. We also show that the category of dual operator modules over a W -algebra generated by a dual operator algebra M , is a subcategory of the category of dual operator modules over M . We develop the theory of the W -dilation which connects the non-selfadjoint dual operator algebra with the W -algebraic framework. In particular we use the maximal W -algebra generated by a dual operator algebra and show that every dual operator module is a w-closed submodule of its ‘maximal dilation’. Indeed, in the case of weak Morita equivalence this maximal dilation turns out to be a W -module. The theory of the W -dilation is a key part of the proof of our main theorem. Date: October 18, 2008.