A Morita theorem for dual operator algebras

We prove that two dual operator algebras are weak∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401–2412] 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. We also develop the theory of the W∗-dilation, which connects the non-selfadjoint dual operator algebra with the W∗-algebraic framework. In the case of weak∗ Morita equivalence, this W∗-dilation is a W∗-module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the W∗-dilation is a key part of the proof of our main theorem.