Given a factor *M* (=von Neumann alg. with center ℂ), let us write *BIM* for the ⊗-*C*\*-category of *M*-*M*-bimodules.



> 
> Which ⊗-*C*\*-categories can one faithfully embed into *BIM*?
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⓵ Are there ***necessary*** conditions for a ⊗-*C*\*-category to be representable in *BIM*?  

⓶ Are there ***sufficient*** conditions for a ⊗-*C*\*-category to be representable in *BIM*?  
  



*Comment:*  

I suspect that a lot is known about ⓶ in relation with the theory of planar algebras...  

I specially care about ⓵: are there examples of ⊗-*C*\*-categories that don't embed?



representing tensor-C*-categories in BIM
I don't know about necessary conditions, but here are some results concerning sufficient conditions:


* In MR1749868, Hayashi and Yamagami realize amenable $C^\*$-tensor categories in the category of bifinite (Jones index) bimodules of the hyperfinite $II\_1$-factor.-
* In arXiv:0811.1764v4, Stefaan Vaes and Sébastien Falguières showed that "the representation category of any compact group is the [bifinite] bimodule category of a $II\_1$-factor," i.e., given a compact group $G$ with representation category $C$, there is a $II\_1$-factor $M$ whose category of bifinite bimodules is exactly $C$.-
* Recently, Sven Raum and Sébastien Falguières showed that "all finite $C^\*$-tensor categories are [bifinite] bimodule categories of a $II\_1$-factor," i.e., given a finite $C^\*$-tensor category $C$, there is a $II\_1$-factor $M$ whose category of bifinite bimodules is exactly $C$. This paper has yet to appear on the arXiv. Here is the [link](http://genco-2011.institut.math.jussieu.fr/program.html) to the conference at which the talk was given.-


I don't know of such results for type $III$ factors.


