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 {$4\cdot cos^2(\pi/n);n=3,4,5,...$} $\cup$ $[4,\infty]$?

<hr>

<b>Remark:</b> 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 <i>Jones index</i>. 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.