# Examples of non-extremal subfactors

Every subfactor $(N \subset M)$ in this post are supposed to be finite index inclusion of ${\rm II}_1$ factors.

Definition (here p64): Such a subfactor is called extremal if $tr_{N'} = tr_M$ on $N' \cap M$ (where $N′$ is calculated on any Hilbert space on which M acts with finite $M$-dimension).

In this paper p25, remark 6.6. states that the $A_{\infty}^{(1)}$ subfactors, realizing every index $\ge 4$, are non-extremal.

Question 1: Why the $A_{\infty}^{(1)}$ subfactors are non-extremal?

Moreover, the $A_{\infty}^{(1)}$ subfactors are maximal, i.e. admits no non-trivial intermediate subfactor (see here).
Question 2: What are the (smallest in index) known examples of non-$A_{\infty}^{(1)}$ non-extremal maximal subfactor?

Remark: An irreducible subfactor (i.e. $N' \cap M = \mathbb{C}$) is a fortiori extremal. The finite depth subfactors are also known to be extremal. So an example should be non-irreducible, infinite depth and maximal (as $A_{\infty}^{(1)}$).
I've added the assumption maximal for avoiding answers like $A_{\infty}^{(1)} \otimes X$.

But any $A_{\infty}^{(1)}$ subfactor admits an intermediate von Neumann algebra (see also here).
Question 3: Is there a non-extremal subfactor without non-trivial intermediate von Neumann algebra?