Realization of rigid C∗-tensor categories via Tomita bimodules

Abstract

Starting from a (small) rigid C∗-tensor category C with simple unit, we construct von Neumann algebras. These algebras are factors of type II or IIIλ,λ∈(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. If the spectrum is infinite we realize the whole tensor category as endomorphisms of these algebras. Furthermore, if the Tomita structure is trivial, the algebras that we get are an amplification of the free group factors with infinitely (possibly uncountably) many generators.