My question is based on the following vague belief, shared by many people: It should be possible to use von Neumann algebras in order to define the cohomology theory TMF (topological modular forms) in the same way one uses Hilbert spaces in order to define topological K-theory. More precisely, one expects hyperfinite type $\mathit{III}$ factors to be the analogs of (separable) infinite dimensional Hilbert spaces.
Now, here is a fundamental difference between Hilbert spaces and type $\mathit{III}$ factors: The category of Hilbert spaces has two monoidal structures: direct sum $\oplus$, and tensor product $\otimes$, and both of them preserve the property of being an infinite dimensional Hilbert space.
In von Neumann algebras, the tensor product of two hyperfinite type $\mathit{III}$ factors is again a hyperfinite type $\mathit{III}$ factor, but their direct sum isn't (it's not a factor).
Hence my question: are there other monoidal structures on the category of von Neumann algebras that I might not be aware of? More broadly phrased, how many ways are there of building a new von Neumann algebra from two given ones, other than tensoring them together?
Ideally, I would need something that distributes over tensor product, and that preserves the property of being a hyperfinite type $\mathit{III}$ factor… but that might be too much to ask for.
Best Answer
The category of von Neumann algebras W* admits a variety of monoidal structures of three distinct flavors.
(1) W* is complete and therefore you have a monoidal structure given by the categorical product.
(2a) W* is cocomplete and therefore you also have a monoidal structure given by the categorical coproduct.
(2b) I suspect that there is also a “spatial coproduct”, just as there is a categorical tensor product and a spatial tensor product (see below). The spatial coproduct should correspond to a certain central projection in the categorical coproduct. Perhaps the spatial coproduct is some sort of coordinate-free version of the free product mentioned in Dmitri Nikshych's answer.
(3a) For any two von Neumann algebras M and N consider the functor F from W* to Set that sends a von Neumann algebra L to the set of all pairs of morphisms M→L and N→L with commuting images. The functor F preserves limits and satisfies the solution set condition, therefore it is representable. The representing object is the categorical tensor product of M and N.
(3b) There is also the classical spatial tensor product. I don't know any good universal property that characterizes it except that there is a canonical map from (3a) to (3b) and its kernel corresponds to some central projection in (3a). Perhaps there is a nice description of this central projection.
Since your monoidal structure is of the third flavor and you don't want a monoidal structure of the first flavor, I suggest that you try a monoidal structures of the second flavor. I suspect that the spatial coproduct of two factors is actually a factor. You are lucky to work with factors, because in the commutative case 2=3, in particular 2a=3a and 2b=3b.