A categorical Connes' $χ(M)$

Abstract

Popa introduced the tensor category χ(M ) of approximately inner, centrally trivial bimodules of a II1 factor M , generalizing Connes' χ(M ). We extend Popa's notions to define the W * -tensor category End loc (C) of local endofunctors on a W * -category C. We construct a unitary braiding on End loc (C), giving a new construction of a braided tensor category associated to an arbitrary W * -category. For the W * -category of finite modules over a II1 factor, this yields a unitary braiding on Popa's χ(M ), which extends Jones' κ invariant for χ(M ).Given a finite depth inclusion M0 ⊆ M1 of non-Gamma II1 factors, we show that the braided unitary tensor category χ(M∞) is equivalent to the Drinfeld center of the standard invariant, where M∞ is the inductive limit of the associated Jones tower. This implies that for any pair of finite depth non-Gamma subfactors N0 ⊆ N1 and M0 ⊆ M1, if the standard invariants are not Morita equivalent, then the inductive limit factors N∞ and M∞ are not stably isomorphic.

A characterization of a finite-dimensional commuting square producing a subfactor of finite depth

“…Also see [19] for a more brief description relevant to subfactor theory (and conformal field theory). A recent paper [7] also explains the framework in a concise manner.…”