representing tensor-C*-categories in BIM

Question

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?

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?

Answer 1

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 to the conference at which the talk was given.-

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