A classification of all finite index subfactors for a class of group-measure space $\mathrm{II}_1$ factors

Abstract

Abstract. We provide a family of group-measure space II 1 factors for which all finite index subfactors can be explicitly listed. In particular, the set of all indices of irreducible subfactors can be computed. Concrete examples show that this index set can be any set of natural numbers that is closed under taking divisors and least common multiples. (2010). Primary 46L37; Secondary 46L36, 28D15. Mathematics Subject Classification

C∗-Tensor Categories and Subfactors for Totally Disconnected Groups

“…We finally note that the rigid C * -tensor categories C 1,f (K < G) and C 3,f (K < G) also arise in a different way as categories of bimodules over a II 1 factor in the case where K < G is the Schlichting completion of a Hecke pair Λ < Γ, cf. [DV10,Section 4].…”

“…In this way, we have defined the rigid C * -tensor category C 4,f (Λ < Γ) consisting of the finite rank objects in C 4 . Note that, in a different context, this rigid C * -tensor category C 4,f (Λ < Γ) already appeared in [DV10,Section 4].…”

Rigidity for von Neumann Algebras and Their Invariants

“…Explicit examples were provided in [Va07] where also several concrete calculations of Bimod M were made. These calculations were exploited in [DV10] to give a full classification of all finite index subfactors of certain II 1 factors. In [FV08] every representation category of a compact group K is realized as Bimod M .…”

TensorC∗-categories arising as bimodule categories of II 1 factors

“…isomorphism classes of finite index bimodules and fusion rules, were calculated. The complete calculation of the category of bimodules over II 1 factors coming from [Vae07] was obtained by Deprez and Vaes in [DV10]. Even more is proven in [DV10], since the C˚-bicategory of II 1 factors commensurable with M , i.e.…”

“…Theorem B. There exists a II 1 factor M such that In [Vae07], [FV08] and [DV10] only categories with at most countably many isomorphism classes of irreducible objects were obtained as bimodule categories of II 1 factors. In this article we give examples of II 1 factors M such that BimodpM q can be calculated and has uncountably many pairwise non isomorphic irreducible objects.…”