# Are there only finitely many maximal irreducible amenable subfactors at fixed finite index?

A subfactor $N \subset M$ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M$.

Question: Are there only finitely many maximal irreducible amenable subfactors at fixed finite index (up to isomorphism)?

Bonus question: let $\alpha$ the index of a irreducible amenable subfactor.
Is there a maximal irreducible amenable subfactor of index $\alpha$ ?

• Presumably, you are imagining fixing the isomorphism class of $N$ and $M$ as von Neumann algebras (otherwise, even for index=1, there are infinitely many). Another way of formalizing the question so that it has a chance of have a positive answer, is to ask for isomorphism classes of planar algebras. Jun 8 '13 at 18:00
• What about subfactors of index 4? I think subfactors of affine type $D_n$ are all maximal. Jun 8 '13 at 18:12
• Dave Penneys says to me that each the $D_{n}^{(1)}$ subfactors decompose into intermediate subfactors isomorphic to $R^{\mathbb{Z}_{2}} \subset R$ (with $R$ the hyperfinite $II_{1}$ factor), but I have not verified this information, I will try to find a reference. Jun 8 '13 at 18:32
• If anyone know a reference containing the explicit intermediate subfactors lattice for all the $D^{(1)}_{n}$ subfactors, please post it here. Jun 10 '13 at 12:54
• It says: "If $\Gamma$ is $D^{(1)}_n$ $n > 4$ or $D^{(1)}_\infty$, then $N\subset M$ has the only one nontrivial intermediate subfactor." in the proof they use that the Galois group is $\mathbb{Z}_2$. Jun 16 '13 at 15:53