Realization of rigid C$^*$-tensor categories via Tomita bimodules

Understand

Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects.

• These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$, depending on whether the spectrum of the category is finite or infinite) or they can be of type III$_\lambda$, $\lambda\in (0,1]$.
• The choice of type is tuned by the choice of Tomita structure (defined in the paper) on certain bimodules we use in the construction.
• Moreover, if the spectrum is infinite we realize the whole tensor category directly as endomorphisms of these algebras, with finite Jones index, by exhibiting a fully faithful unitary tensor functor $F:\mathscr{C} \hookrightarrow End_0(\Phi)$ where $\Phi$ is a factor (of type II or III).