A W*-category is a horizontal categorification of a von Neumann algebra.
W*-categories are in the same relation to C*-categories as von Neumann algebras are to C*-algebras.
A W*-category is a C*-category such that for any objects , the hom-object admits a predual as a Banach space. That is, there is a Banach space such that is isomorphic to in the category of Banach spaces and contractive maps (alias short maps).
A W*-functor is a functor such that for any morphism in and the map of Banach spaces admits a predual for any objects and in .
The good notion of natural transformations between W*-categories is given by bounded natural transformations: a natural transformation between W*-functors is bounded if the norm of is bounded with respect to the object of .
W*-categories, W*-functors, and bounded natural transformations form a bicategory.
This bicategory is a good setting to work with objects like Hilbert spaces, Hilbert W*-modules over von Neumann algebras, W*-representations of von Neumann algebras, etc.
In particular, in this bicategory, the category of Hilbert spaces has infinite direct sums (generalizing the definition of a biproduct to infinite families of objects), unlike in the usual bicategory of categories, functors, and natural transformations, where it only has finite limits and finite colimits.
The same is true for Hilbert W*-modules over von Neumann algebras, W*-representations of von Neumann algebras.
Last revised on April 25, 2021 at 03:07:05. See the history of this page for a list of all contributions to it.