Department of Mathematics

For a finite von Neumann algebra $M$ with separable predual, we construct a braiding on the unitary tensor category $\tilde{\chi}(M)$ of dualizable approximately inner and centrally trivial $M$-$M$ bimodules, generalizing the usual notions for automorphisms and extending Connes’

$\chi(M)$. Our unitary braiding on $\tilde{\chi}(M)$ extends Jones’ $\kappa$ invariant. 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 inclusions $N_{0}\subseteq N_{1}$ 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 Morita equivalent.  This talk is based on joint work with Quan Chen and David Penneys.