A categorical Connes' $\chi(M)$

TL;DR

This paper extends Popa's tensor category to a braided tensor category for W*-categories, linking it to Jones' invariants and the Drinfeld center, with implications for classifying subfactors via their standard invariants.

Contribution

It introduces a new braided tensor category structure on Popa's tensor category, extending Jones' invariants, and relates it to the Drinfeld center of the standard invariant for subfactors.

Findings

Constructed a unitary braiding on the endofunctor category of a W*-category.

Extended Jones' $ ext{kappa}$ invariant to Popa's $ ilde{ ext{chi}}(M)$.

Established non-isomorphism criteria for inductive limit factors based on standard invariants.

Abstract

Popa introduced the tensor category $\tilde{\chi}(M)$ of approximately inner, centrally trivial bimodules of a $\rm{II}_{1}$ factor $M$, generalizing Connes' $\chi(M)$. We extend Popa's notions to define the $\rm W^*$-tensor category $\operatorname{End}_{\rm loc}(\mathcal{C})$ of local endofunctors on a $\rm W^*$-category $\mathcal{C}$. We construct a unitary braiding on $\operatorname{End}_{\rm loc}(\mathcal{C})$, giving a new construction of a braided tensor category associated to an arbitrary $\rm W^*$-category. For the $\rm W^*$-category of finite modules over a $\rm{II}_{1}$ 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 $\rm{II}_1$ 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.