A categorical Connes' $\chi (M)$

Abstract

Popa introduced the tensor category $\tilde{\chi }(M)$ of approximately inner, centrally trivial bimodules of a ${\mathrm{II}}_1$ factor $M$, generalizing Connes' $\chi (M)$. We extend Popa's notions to define the ${\mathrm{W}}^{\ast }$-tensor category ${\mathrm{End}}_{\mathrm{loc}}(\mathcal{C})$ of local endofunctors on a ${\mathrm{W}}^{\ast }$-category $\mathcal{C}$. We construct a unitary braiding on ${\mathrm{End}}_{\mathrm{loc}}(\mathcal{C})$, giving a new construction of a braided tensor category associated to an arbitrary ${\mathrm{W}}^{\ast }$-category. For the ${\mathrm{W}}^{\ast }$-category of finite modules over a ${\mathrm{II}}_1$ factor, this yields a unitary braiding on Popa's $\tilde{\chi }(M)$, which extends Jones' $\kappa$ invariant for $\chi (M)$. Given a finite depth inclusion $M_0\subseteq M_1$ of non-Gamma ${\mathrm{II}}_1$ factors, we show that the braided unitary tensor category $\tilde{\chi }(M_{\infty })$ is equivalent to the Drinfeld center of the standard invariant, where $M_{\infty }$ is the inductive limit of the associated Jones tower. This implies that for any pair of finite depth non-Gamma subfactors $N_0\subseteq N_1$ and $M_0\subseteq M_1$, if the standard invariants are not Morita equivalent, then the inductive limit factors $N_{\infty }$ and $M_{\infty }$ are not stably isomorphic.

arXiv comment52 pages, many tikz figures

References in corpus (12)

• Multi-interval Subfactors and Modularity of Representations in Conformal Field Theory
• Random matrices, free probability, planar algebras and subfactors
• Local fields in boundary conformal QFT
• Traces on Module Categories over Fusion Categories
• Haag duality for Kitaev's quantum double model for abelian groups
• Q-system completion for C* 2-categories
• A semi-finite algebra associated to a planar algebra
• A planar calculus for infinite index subfactors
• Subfactors, Planar Algebras and Rotations
• Categorically Morita equivalent compact quantum groups
• On the classification of inductive limits of II${}_1$ factors with spectral gap
• Unitary dual functors for unitary multitensor categories