C*-groupoides quantiques et inclusions de facteurs : Structure symetrique et autodualite, action sur le facteur hyperfini de type II1

TL;DR

This paper introduces a new symmetric duality for quantum C*-groupoids associated with finite index type II1 factor inclusions, showing their actions on the hyperfinite factor and proposing deformations to regular structures.

Contribution

It defines a symmetric duality for quantum C*-groupoids, demonstrates their outer action on the hyperfinite II1 factor, and constructs deformations to regular quantum C*-groupoids.

Findings

Temperley-Lieb algebras are selfdual quantum C*-groupoids

Finite quantum C*-groupoids act outerly on the hyperfinite II1 factor

Deformation to regular quantum C*-groupoids is possible

Abstract

Let N_0 \subset N_1 a depth 2, finite index inclusion of type II1 factors and N_0 \subset N_1 \subset N_2 \subset N_3 ... the corresponding Jones tower. D. Nikshych et 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 symetric 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 choosen selfdual. I show that every finite-dimensional connexe quantum C*-groupoid acts outerly on the type II1 hyperfinite factor. In the light of this particular case, I propose a deformation of any finite quantum C*-groupoid to an regular finite quantum C*-groupoid. In the appendix, a new construction of the factors on which two dual regular finite quantum C*-groupoids act is given. The finite quantum C*-groupoids obtained from the built tower are isomorphic to the initial ones.