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

2025/11/25 by Biplab Pal, Pal, Biplab
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2511.19954

Abstract

Let B ⊂ 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' ∩ A₁ 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 A₁ is isomorphic to the crossed product A \rtimes (B' ∩ A₁). 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 II₁ factor setting.

Citations

• Higher reflections and entropy of canonical shifts for inclusions of C^*-algebras with finite Watatani index
• Regular inclusions of simple unital C^*-algebras
• Inclusions of simple C^*-algebras arising from compact group actions
• Orthogonal Unitary Bases and a Subfactor Conjecture
• Inclusions of C^*-algebras arising from fixed-point algebras
• A few remarks on Pimsner-Popa bases and regular subfactors of depth 2
• Cancellation for inclusions of C*-algebras of finite depth
• Frobenius extensions and weak Hopf algebras
• Weak Hopf algebra symmetries of C^*-algebra inclusions
• Inclusions of von Neumann Algebras and Quantum groupoids III
• Duality for actions of weak Kac algebras and crossed product inclusions of II₁ factors
• Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers
• Weak Hopf Algebras I: Integral Theory and C^*-structure

Related

• Weak Hopf algebra symmetries of C^*-algebra inclusions
• A characterization of depth 2 subfactors of II₁ factors
• Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers
• Regular inclusions of simple unital C^*-algebras
• Inclusions of simple C^*-algebras