A categorical Connes' $\chi(M)$

Author name not available (Why is that?)

Publication date: 11 November 2021

Abstract: Popa introduced the tensor category

i l d e c h i (M)

of approximately inner, centrally trivial bimodules of a

m {I I}_{1}

factor

M

, generalizing Connes'

c h i (M)

. We extend Popa's notions to define the

m W^{*}

-tensor category

o p e r a t o r n a m e {E n d}_{m l o c} (m a t h c a l C)

of local endofunctors on a

m W^{*}

-category

m a t h c a l C

. We construct a unitary braiding on

o p e r a t o r n a m e {E n d}_{m l o c} (m a t h c a l C)

, giving a new construction of a braided tensor category associated to an arbitrary

m W^{*}

-category. For the

m W^{*}

-category of finite modules over a

m {I I}_{1}

factor, this yields a unitary braiding on Popa's

i l d e c h i (M)

, which extends Jones'

k a p p a

invariant for

c h i (M)

. Given a finite depth inclusion

M_{0} s u b s e t e q M_{1}

of non-Gamma

m {I I}_{1}

factors, we show that the braided unitary tensor category

i l d e c h i (M_{i n f t y})

is equivalent to the Drinfeld center of the standard invariant, where

M_{i n f t y}

is the inductive limit of the associated Jones tower. This implies that for any pair of finite depth non-Gamma subfactors

N_{0} s u b s e t e q N_{1}

and

M_{0} s u b s e t e q M_{1}

, if the standard invariants are not Morita equivalent, then the inductive limit factors

N_{i n f t y}

and

M_{i n f t y}

are not stably isomorphic.

No records found.

This page was built for publication: A categorical Connes' $\chi(M)$