By definition, a von Neumann algebra is a C*‑algebra A that admits a predual, i.e., a Banach space Z such that Z* is isomorphic to the underlying Banach space of A. (We require that isomorphisms in the category of Banach spaces preserve the norm.)

Moreover, a morphism of von Neumann algebras is a morphism f: A→B of the underlying C*‑algebras that admits a predual, i.e., a morphism of Banach spaces g: Y→Z such that g*: Z*→Y* is isomorphic to f in the category of morphisms of Banach spaces.

A theorem by Sakai states that all preduals of A induce the same weak topology on A (the ultraweak topology). In particular, every predual is canonically isomorphic to the dual of A in the ultraweak topology and the predual is unique up to unique isomorphism.

The same is true for morphisms: The unique predual of a morphism f: A→B of von Neumann algebras is its dual f*: B*→A* in the ultraweak topology. A morphism of the underlying C*‑algebras has a predual if and only if it is continuous in the ultraweak topology.

Thus the predual can be seen as a functor L_{1} that sends
a von Neumann algebra to its dual in the ultraweak topology
and likewise for morphisms. The functor L_{1} is faithful,
which boils down to the fact that an element of a von Neumann
algebra that vanishes on every element of the predual
must be zero.

Whenever we have a faithful functor F: C→D it is natural to try to factor it as a composition of two functors G: C→E and H: E→D, where the objects of E are the objects of D equipped with some additional structures, the morphisms of E are the morphisms of D that preserve these structures, H is the functor that forgets these structures, and G is an equivalence of categories. In our case we are looking for additional structures on the predual such that morphisms of preduals that preserve these structures are precisely morphisms that come from morphisms of von Neumann algebras. Since the functor G is an equivalence of categories, this amounts to an alternative definition of von Neumann algebras in terms of preduals.

Two such structures on the predual are easy to identify.
The first one is given by dualizing the unit map 1: k→A.
Here k is the field of scalars over which all von Neumann
algebras and Banach spaces are defined.
The dual map is the famous Haagerup trace tr: L_{1}(A)→k.

The second structure is given by dualizing the conjugate-linear involution *: A→A.
The dual is the modular conjugation *: L_{1}(A)→L_{1}(A),
which is important in Tomita-Takesaki theory.
Finally, the trace commutes with the conjugation.

All morphisms of preduals that come from morphisms of von Neumann algebras preserve these two structures. Dualizing all morphisms of preduals that preserve these two structures gives us ultraweakly continuous maps of the original von Neumann algebras that preserve the unit and the involution.

However, there is no guarantee that these maps preserve the multiplication,
at least I am not aware of any proof or refutation of this statement.
We can naïvely try to dualize the multiplication map A⊗A→A
and get a map of the form L_{1}(A)→L_{1}(A)⊗L_{1}(A).
However, it is unclear what kinds of tensor products we should use
(see a question on this matter)
and whether dualizing A⊗A actually gives us L_{1}(A)⊗L_{1}(A).

**Question 1:** Is there an example of a map of preduals such that
its dual map preserves the unit and the involution but does not
preserve the multiplication?
If yes, what kind of additional structure
maps between preduals should preserve to ensure that they
actually come from morphisms of von Neumann algebras?
In particular, can we dualize the product on a von Neumann algebra using some kind of tensor product?

**Question 2:** What Banach spaces (possibly equipped with a trace,
an involution, and perhaps some other structures)
arise as preduals of von Neumann algebras?