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

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.

💡 Research Summary

The paper investigates inclusions of simple unital C*-algebras B⊂A equipped with a finite Watatani index conditional expectation, focusing on the case where the inclusion has depth 2. Starting from the Watatani index theory, the authors construct the tower of basic constructions B⊂A⊂A₁⊂A₂⊂⋯, each step accompanied by a Jones projection eₙ and a unique minimal conditional expectation. The relative commutants B′∩Aₙ are shown to be finite‑dimensional and carry faithful tracial states obtained by iterating the conditional expectations.

A central achievement is the establishment of a non‑degenerate duality between the second relative commutant P:=B′∩A₁ and the next‑level commutant Q:=A′∩A₂. The pairing ⟨x,w⟩=d⁻¹τ⁻² tr(x e₂ e₁ w) (with d a normalization constant) is proved to be perfect, allowing the authors to transport algebraic structures from one side to the other. Using Fourier transforms and rotation operators defined in the C*-setting, they construct a comultiplication Δ, counit ε, and antipode S on Q, turning Q into a coalgebra that is dual to the algebra P.

When the inclusion has depth 2, the equality (B′∩A₁) e₂ (B′∩A₁)=B′∩A₂ holds, which implies that Q is exactly the coalgebra dual of P. Consequently, the authors endow P with the structure of a weak C*-Hopf algebra: Δ is a *‑preserving algebra homomorphism, ε satisfies the weak counit axioms, and S fulfills the antipode relations characteristic of weak Hopf algebras. The target and source counital maps ε_t and ε_s define Cartan subalgebras P_t and P_s, and the paper discusses conditions under which P is connected or biconnected.

Having identified P as a weak C*-Hopf algebra, the authors construct a left action of P on the original algebra A. The action is given explicitly by
 y ▹ a = τ⁻¹ ∑_i λ_i E₁(e₁ y λ_i* a),
where {λ_i} is a quasi‑basis for the minimal expectation E₀. This action satisfies the compatibility conditions with Δ, ε_t, and the *‑structure, and the fixed‑point subalgebra A^P = {a∈A | y▹a = ε_t(y)·a ∀y∈P} is proved to coincide exactly with B.

The crossed product A⋊P is then defined as the relative tensor product A⊗_{P_t}P, equipped with the multiplication