A Morita theorem for dual operator algebras
arXiv:0806.2704
Abstract
We prove that two dual operator algebras are weak Morita equivalent 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 -dilation, which connects the non-selfadjoint dual operator algebra with the -algebraic framework. In the case of weak Morita equivalence, this -dilation is a -module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the -dilation is a key part of the proof of our main theorem.
21 pages