A Morita theorem for dual operator algebras

TL;DR

This paper establishes a Morita equivalence framework for dual operator algebras, linking their module categories and introducing a $W^*$-dilation theory to connect non-selfadjoint and $W^*$-algebraic structures.

Contribution

It provides a characterization of weak$^*$ Morita equivalence for dual operator algebras using module categories and tensor products, extending Morita theory to the dual operator algebra setting.

Findings

Characterization of weak$^*$ Morita equivalence via module categories

Development of $W^*$-dilation connecting dual operator algebras to $W^*$-algebras

Identification of the $W^*$-dilation as a $W^*$-module over a von Neumann algebra

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 $W^*$-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.