Closedness of index values for subfactors

Abstract:

Let $\left \{ {{A_i} \subset {B_i}} \right \}$ be a collection of inclusions of finite factors with the indices $\left \{ {\left [ {{B_i}:{A_i}} \right ]} \right \}$ bounded and each with trivial relative commutant. Then any ${W^ * }$ ultraproduct yields an inclusion with trivial relative commutant whose index is the ultralimit of $\left \{ {\left [ {{B_i}:{A_i}} \right ]} \right \}$. In particular, the set of values of indices arising from pairs of factors with trivial relative commutant is a closed subset of ${{\mathbf {R}}^ + }$.