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

Marie-Claude David|arXiv (Cornell University)|Jun 26, 2003

Advanced Operator Algebra Research18 references3 citations

TL;DR

This paper introduces a symmetric duality for quantum C*-groupoids associated with finite depth, finite index inclusions of type II₁ factors, eliminating the need for new involution definitions. It proves that all finite-dimensional connected quantum C*-groupoids act outerly on the hyperfinite II₁ factor and proposes a deformation to regular quantum C*-groupoids, with Temperley-Lieb algebras shown to be self-dual under this framework.

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.

Motivation & Objective

• To resolve asymmetry in existing duality constructions for quantum C*-groupoids arising from subfactors.
• To establish a symmetric duality between relative commutants in Jones towers without altering the involution.
• To prove that every finite-dimensional connected quantum C*-groupoid acts outerly on the hyperfinite II₁ factor.
• To propose a deformation procedure transforming any finite quantum C*-groupoid into a regular one.
• To demonstrate that Temperley-Lieb algebras are self-dual quantum C*-groupoids under the new duality framework.

Proposed method

• Introduces a new duality structure between relative commutants $N'_0 \cap N_2$ and $N'_1 \cap N_3$ in a Jones tower of type II₁ factors.
• Uses the trace $tr$ on $N_1$ and a modified pairing $\langle a,b\rangle = [N_1:N_0]^2 \, tr(af_2f_1Hb)$ to define the duality without changing the involution.
• Applies the theory of weak Hopf algebras and Kac algebras to define co-product, co-unit, and antipode structures on the groupoids.
• Constructs a crossed product $M_1 \rtimes A$ to realize the action of a quantum groupoid $A$ on a factor $M_1$.
• Utilizes the modular automorphism and conditional expectations to define Haar measures and traces on the groupoid algebra.
• Proposes a deformation of a finite quantum C*-groupoid $A$ via a new dual structure $B.A$, leading to a regularized version.

Experimental results

Research questions

• RQ1 Can a symmetric duality be defined for quantum C*-groupoids associated with subfactors without altering the involution?
• RQ2 Are Temperley-Lieb algebras self-dual as quantum C*-groupoids under the new duality framework?
• RQ3 Does every finite-dimensional connected quantum C*-groupoid admit an outer action on the hyperfinite II₁ factor?
• RQ4 Can any finite quantum C*-groupoid be deformed into a regular one via a new duality construction?
• RQ5 How do the co-product, co-unit, and antipode structures behave under the new duality in the context of Jones towers?

Key findings

• The new duality allows a symmetric construction of quantum C*-groupoids on $N'_0 \cap N_2$ and $N'_1 \cap N_3$ without modifying the involution.
• Temperley-Lieb algebras are shown to be self-dual quantum C*-groupoids under the proposed duality.
• Every finite-dimensional connected quantum C*-groupoid acts outerly on the hyperfinite II₁ factor.
• A deformation procedure is constructed that transforms any finite quantum C*-groupoid into a regular one via a dual structure.
• The relative commutants in the Jones tower inherit the original quantum C*-groupoid structures under the new duality, ensuring consistency.
• The construction yields a Jones tower of hyperfinite II₁ factors where the dual groupoids act via crossed products, preserving the algebraic structure.