Realizations of algebra objects and discrete subfactors

2017/04/06 by Jones, Corey, Penneys, David · 3 citations
#18D10 #46L37 #Category Theory (math.CT) #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1704.02035

Abstract

We give a characterization of extremal irreducible discrete subfactors (N⊆ M, E) where N is type \rm II₁ in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal N-N bilinear ucp maps which preserve the state τ∘ E, and the morphisms for W*-algebra objects are categorical ucp morphisms. As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our equivalence provides many new examples of discrete inclusions (N⊆ M, E), in particular, examples where M is type \rm III coming from non Kac-type discrete quantum groups and associated module W*-categories. Finally, we obtain a Galois correspondence between intermediate subfactors of an extremal irreducible discrete inclusion and intermediate W*-algebra objects.

Cited by

• Properly Outer Actions of Tensor Categories on C^*-algebras
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Related

• Discrete Inclusions of C*-algebras
• Inclusions of Operator Algebras from Tensor Categories: beyond irreducibility
• Q-systems and compact W*-algebra objects
• Remarks on C*-discrete inclusions
• Subfactors associated to compact Kac algebras