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 {$4\cdot \cos^2(\pi/n);n=3,4,5,...$} $\cup$ $[4,\infty]$? In other words, given a real number in the above set, is there a subfactor whose minimal index is that real number?

**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.

Jones' index is known to take all the above values. But the subfactors used in the construction are not irreducible (and one can also check that their minimal index is different from their Jones index).