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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