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 intoBIM?

⓵ Are there ** necessary** conditions for a ⊗-

*C**-category to be representable in

*BIM*?

⓶ Are there

**conditions for a ⊗-**

*sufficient**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?