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# Journal of Operator Theory

Volume 54, Issue 1, Summer 2005  pp. 27-68.

$C$*-groupo\''ides quantiques et inclusions de facteurs Structure sym\'etrique et autodualit\'e, action sur le facteur hyperfini de type $\mathrm{II}_{1}$

Authors Marie-Claude David

Summary:  Let $N_{0} \subset N_{1}$ a depth $2$, finite index inclusion of type $\mathrm{II}_{1}$ factors and $N_0 \subset N_1 \subset N_2 \subset N_3 \subset\cdots$ the corresponding Jones tower. D. Nikshych and L. Vainerman built dual structures of quantum $C^*$-groupoid on the relative commutants $N'_{0} \cap N_{2}$ et $N'_{1} \cap N_{3}$. Here I define a new duality which allows a symmetric construction without changing the involution. So the Temperley-Lieb algebras are selfdual quantum $C^*$-groupoids and the quantum $C^*$-groupoids associated to a finite depth finite index inclusion can be chosen selfdual. I show that every finite-dimensional connected quantum $C^*$-groupoid acts outerly on the type $\mathrm{II}_{1}$ hyperfinite factor. In light of this particular case, I propose a deformation of any finite quantum $C^*$-groupoid to a regular finite quantum $C^*$-groupoid.

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