Some endomorphisms of the hyperfinite $II_1$ factor
  For any finite dimensional C*-algebra A with any trace vector {\vec s} whose components are rational numbers, we give an endomorphism {\Phi} of the hyperfinite II_1 factor R such that: forall k in {\mathbb N} {\Phi}^k (R)' \cap R= \otimes^k A The canonical trace {\tau} on R extends the trace vector {\vec s} on A. As a corollary, we construct a one-parameter family of inclusions of hyperfinite II_1 factors N^{\lambda} \subset M^{\lambda} with trivial relative commutant (N^{\lambda})' \cap M^{\lambda}= {\mathbb C} and with the Jones index [M^{\lambda}: N^{\lambda}]= \lambda^{-1} \in (4, \infty) \cap {\mathbb Q} This partially solves the problem of finding all possible values of indices of subfactors with trivial relative commutant in the hyperfinite II_1 factor, by showing that any rational number \lambda^{-1} > 4 can occur. 
Hsiang-Ping Huang