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Journal of Operator Theory

Volume 81, Issue 2, Spring 2019  pp. 433-479.

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

Authors:  Luca Giorgetti (1), Wei Yuan (2)
Author institution:(1) Dipartimento di Matematica, Universita di Roma Tor Vergata, Via della Ricerca Scientifica, 1, Roma, I-00133, Italy and Dipartimento di Matematica ``Guido Castelnuovo'', Sapienza Universita di Roma, Piazzale Aldo Moro, 5, Roma, I-00185, Italy
(2) Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China \textit{and} School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

Summary: Starting from a (small) rigid $C^*$-tensor category $\mathscr C$ with simple unit, we construct von Neumann algebras. These algebras are factors of type II or 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. 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.

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

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