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.
DOI: http://dx.doi.org/10.7900/jot.2018mar08.2219
Keywords: $C^*$-tensor category, pre-Hilbert $C^*$-bimodule, full Fock space construction, free group factor
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