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 L1 that sends
a von Neumann algebra to its dual in the ultraweak topology
and likewise for morphisms. The functor L1 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: L1(A)→k.
The second structure is given by dualizing the conjugate-linear involution *: A→A.
The dual is the modular conjugation *: L1(A)→L1(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 L1(A)→L1(A)⊗L1(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 L1(A)⊗L1(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?
Best Answer
The answer to this is explored in a number of papers. AFAIK, it was first considered by Quigg in "Approximately periodic functionals on C*-algebras and von Neumann algebras.", http://www.ams.org/mathscinet-getitem?mr=806641 Here he considered the Banach space projective tensor product, but this only works for subhomogeneous algebras.
The general case was answered by Effros and Ruan in "Operator space tensor products and Hopf convolution algebras", http://www.ams.org/mathscinet-getitem?mr=2015023 If $M$ is a von Neumann algebra with predual $M_*$ then there is a "tensor product", called the Extended Haagerup Tensor Product $M_* \otimes_{eh} M_*$, and a (complete) contraction $\delta:M_* \rightarrow M_* \otimes_{eh} M_*$ which is the predual (in some slightly technical sense) of the multiplication map $M\otimes M\rightarrow M$. (My scare quotes are because the algebraic tensor product $M_*\otimes M_*$ is not (norm) dense in $M_* \otimes_{eh} M_*$).