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?
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?