A bicategorical approach to Morita equivalence for rings and von Neumann algebras

Comonoidal W*-Morita equivalence for von Neumann bialgebras

A theory of Galois co-objects for von Neumann bialgebras is introduced. This concept is closely related to the notion of comonoidal W*-Morita equivalence between von Neumann bialgebras, which is a Morita equivalence taking the comultiplication structure into account. We show that the property of `being a von Neumann algebraic quantum group' (i.e. `having invariant weights') is preserved under this equivalence relation. We also introduce the notion of a projective corepresentation for a von Neumann bialgebra, and show how it leads to a construction method for Galois co-objects and comonoidal W*…

Bicategories of operator algebras and Poisson manifolds

It is well known that rings are the objects of a bicategory, whose arrows are bimodules, composed through the bimodule tensor product. We give an analogous bicategorical description of C*-algebras, von Neumann algebras, Lie groupoids, symplectic groupoids, and Poisson manifolds. The upshot is that known definitions of Morita equivalence for any of these cases amount to isomorphism of objects in the pertinent bicategory.

动力冯·诺依曼代数的Morita理论

Morita theory for dynamical von Neumann algebras

本文研究动力冯·诺依曼代数的等变Morita等价，证明其保持顺从性等动力性质，并用于改进已有结果、回答关于余顺从性与相对顺从性等价的问题，同时探讨紧情形下与C*-代数Morita等价的关系。

Functoriality and Morita equivalence of operator algebras and Poisson manifolds associated to groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W*(G), and that a Lie groupoid G canonically defines both a C*-algebra C*(G) and a Poisson manifold A*(G). We show that the maps G -> W*(G), G -> C*(G) and G -> A*(G) are functorial with respect to suitable categories. In these categories Morita equivalence is isomorphism of objects, so that these maps preserve Morita equivalence.

A Morita theorem for dual operator algebras

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 w…

动力 von Neumann 代数的 Morita 等价的范畴论解释

A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

本文通过构造表示范畴与对应范畴之间的典范函子，证明了动力 von Neumann 代数的等变 Morita 等价等价于其表示范畴作为模 $W^*$-范畴的等价，并在紧量子群情形下得到范畴等价，给出 Eilenberg-Watts 定理的等变版本。