Given a subfactor $N\to M$ and a conditional expectation $E:M\to N$, there is a numerical invariant $Ind(E)$ associated to to this situation, called the index of $E$. The possible values of $Ind(E)$ are restricted to the set {$4\cdot cos^2(\pi/n);n=3,4,5,...$} $\cup$ $[4,\infty]$.

The minimal conditional expectation is the one that minimizes the value of $Ind(E)$. The minimal index of the subfactor is then defined to be the index of its minimal conditional expectation.

Can the minimal index take all values in {$4\cdot cos^2(\pi/n);n=3,4,5,...$} $\cup$ $[4,\infty]$?

Remark: If the factors are of type $II_1$, there is another preferred conditional expectation: the one that is compatible with the traces. The corresponding index is called the Jones index. This is not the index I care about. Jones' index agrees with the minimal index in the case of irreducible subfactors, but not in general.