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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 24th 2021

    Idea

    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.

    Definition

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

    W*-functors

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

    Bounded natural transformation

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

    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 is both complete and cocomplete, 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.

    References

    v1, current

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