Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries
Biplab Pal
TL;DR
The paper extends the depth-2 duality from II$_1$ factor theory to inclusions of simple unital $C^*$-algebras with finite Watatani index, proving that the second relative commutant $B'\cap A_1$ naturally forms a weak $C^*$-Hopf algebra and acts on $A$ with fixed-point algebra $B$, while $A_1$ is isomorphic to the crossed product $A\rtimes (B'\cap A_1)$. Using a nondegenerate duality between $P=B'\cap A_1$ and $Q=A'\cap A_2$, the authors construct coalgebra structures on the relative commutants and obtain a dual Hopf-symmetric framework that generalizes the Ocneanu–Nikshych–Vainerman theory beyond II$_1$ factors. They further show that, in depth-2, the Hopf structure on $Q$ (and its dual on $P$) can be refined to a weak Kac algebra under suitable index scalar conditions, with a deformed version providing a $*$-structure. The action-crossed-product results include $A_2\cong A_1\rtimes Q$ and $A_1\cong A\rtimes P$, and under irreducibility $P$ becomes a finite-dimensional Kac algebra, yielding a robust, $C^*$-algebraic counterpart to depth-2 duality. Overall, the paper broadens the symmetry-duality paradigm from factor theory to a broader class of simple $C^*$-algebras, with concrete constructions of Hopf actions and crossed products.
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.