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

December 26, 2017

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).
The construction relies on methods from free probability (full Fock space,
amalgamated free products), it does not depend on amenability assumptions, and
it can be applied to categories with uncountable spectrum (hence it provides an
alternative answer to a conjecture of Yamagami \cite{Y3}). Even in the case of
uncountably generated categories, we can refine the previous equivalence to
obtain realizations on $\sigma$-finite factors as endomorphisms (in the type
III case) and as bimodules (in the type II case).
In the case of trivial Tomita structure, we recover the same algebra obtained
in \cite{PopaS} and \cite{AMD}, namely the (countably generated) free group
factor $L(F_\infty)$ if the given category has denumerable spectrum, while we
get the free group factor with uncountably many generators if the spectrum is
infinite and non-denumerable.

Keywords:

C$^*$-tensor category, pre-Hilbert C$^*$-bimodule, full Fock space construction, free group factor