TITLE: Monoidal structures on von Neumann algebras
QUESTION [18 upvotes]: 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.

REPLY [2 votes]: Suppose that there was such a category. Then, all objects would isomorphic to $R$ anyway and the question is how sum and tensor product act on the morphisms. The natural choice of morphisms is the set of endomorphisms of the hyperfinite factor. Now, pick a unital inclusion $\iota :R \oplus R \subset R$ and and isomorphism $\mu \colon R \cong R \otimes R$ and define
$$(\phi \oplus \psi)(x) := \iota(\phi(x) \oplus \psi(x)) \quad \mbox{and} \quad
(\phi \otimes \psi)(x) := \mu( (\phi \otimes \psi)(\mu^{-1}(x))).$$
The question is now how sum and tensor product behave. This of course requires a bit of work (and may depend on the choice of $\mu$ and $\iota$). 
However, let us look more concretely. Assume for the moment that $R = \otimes_{n \in {\mathbb N}} M_2 {\mathbb C}$ (with some fixed state on $M_2 {\mathbb C}$). Then, it seems that the inclusion ${\mathbb C}^{\oplus 2} \subset M_2 {\mathbb C}$ in the first factor gives some $\iota$ and a bijection between ${\mathbb N}$ and ${\mathbb N} \times {\mathbb N}$ yields some map $\mu$. It seems straightforward that this should give a bimonoidal category with the required distributivity isomorphisms.
Added August 15 to incorporate Andre's remarks:
The choice of $\iota$ is basically a choice of an infinite projection $p \in R$ and isomorphisms $\phi : R \to pRp$ and $\psi : R \to (1-p)R(1-p)$. Since all projections in $R$ are equivalent (we are in type $III$) i.e. $p \sim 1-p$, $\phi(p) \sim p$ and $\psi(p) \sim p$ the sum should be coherently associative. I think that it is a first step to provide an (inner) isomorphism of $R$ which maps the triple $p, \psi(p),\psi(1-p)$ to $\phi(p),\phi(1-p),1-p$. I have to really think about it, but each triple is a triple of pairwise orthogonal projections in $R$ which sums up to one. I think, that up to unitary conjugacy, there is only one such triple (in the type $III$-setting).