nLab W*-category

Idea

The concept of $W^\ast$-categories is the special case of that of $C^\ast$-categories like von Neumann algebras (aka $W^\ast$-algebras) are a special case of $C^\ast$-algebras. Hence a more systematic name for $W^\ast$-categories would be $W^\ast$-algebroids or von Neumann algebroids.

Definition

A W*-category is a C*-category $C$ such that for any objects $A,B\in C$, the hom-object $Hom(A,B)$ admits a predual as a Banach space. That is, there is a Banach space $Hom(A,B)_*$ such that $(Hom(A,B)_*)^*$ is isomorphic to $Hom(A,B)$ in the category of Banach spaces and contractive maps (alias short maps).

W*-functors

A W*-functor is a functor $F\colon C\to D$ such that $F(f^*)=F(f)^*$ for any morphism $f$ in $C$ and the map of Banach spaces $Hom(A,B)\to Hom(F(A),F(B))$ admits a predual for any objects $A$ and $B$ in $C$.

Bounded natural transformation

The good notion of natural transformations between W*-categories is given by bounded natural transformations: a natural transformation $t\colon F\to G$ between W*-functors is bounded if the norm of $t_X\colon F(X)\to G(X)$ is bounded with respect to the object $X$ of $C$.

The bicategory of W*-categories

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.

Related concepts

• von Neumann algebra

• $C^\ast$-category

References

• P. Ghez, Ricardo Lima, John E. Roberts, W*-categories, Pacific Journal of Mathematics 120:1 (1985), 79–109 [doi:10.2140/pjm.1985.120.79]

Understanding complete $W^\ast$-categories as 2-Hilbert spaces:

• André Henriques, Nivedita, David Penneys: Complete $W^\ast$-categories [arXiv2411.01678]

Exposition:

• Nivedita: Towards Models for 2-Hilb and 3-Hilb as targets for functorial field theories, talk notes (2024) [pdf]