Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries

TL;DR

This paper establishes a duality between depth 2 inclusions of simple unital $C^*$-algebras and weak $C^*$-Hopf algebra symmetries, extending subfactor theory beyond type II_1 factors.

Contribution

It constructs a weak $C^*$-Hopf algebra from a depth 2 inclusion and shows the algebra acts on the larger algebra with the smaller algebra as fixed points, extending existing duality theories.

Findings

The second relative commutant has a natural weak $C^*$-Hopf algebra structure.

The larger algebra is a crossed product of the smaller algebra by this weak Hopf algebra.

The duality theory extends beyond the $II_1$ factor setting.

Abstract

Let $B \subset A$ be a depth $2$ inclusion of simple unital $C^*$-algebras with a conditional expectation of index-finite type. We show that the second relative commutant $B' \cap A_1$ carries a canonical structure of a weak $C^*$-Hopf algebra. Furthermore, we construct an action of this weak $C^*$-Hopf algebra on $A$ for which $B$ is precisely the fixed-point subalgebra, and we prove that the first basic construction $A_1$ is isomorphic to the crossed product $A \rtimes (B' \cap A_1)$. This provides a $C^*$-algebraic counterpart of the duality between depth $2$ subfactors and weak Hopf algebra symmetry, extending the Ocneanu-Nikshych-Vainerman theory beyond the $II_1$ factor setting.