Iterating the basic construction

Abstract:

Let $N \subset M$ be a pair of type II$_{1}$ factors with finite Jones’ index and $N \subset M \subset {M_1} \subset {M_2} \subset \cdots \subset {M_n} \subset \cdots \subset {M_{2n + 1}}$ be the associated tower of type II$_{1}$ factors obtained by iterating Jones’ basic construction. We give an explicit formula of a projection in ${M_{2n + 1}}$ which implements the conditional expectation of ${M_n}$ onto $N$, thus showing that ${M_{2n + 1}}$ comes naturally from the basic construction associated to the pair $N \subset {M_n}$. From this we deduce several properties of the relative commutant $N’ \cap {M_n}$.