Constructing a non-degenerate commuting square from equivalent systems of bimodules

When we have two inclusions of AFD II$_1$ factors $A \subset B$ and $C \subset D$ with finite index and finite depth and the system of the $B$-$B$ bimodules is equivalent to that of the $C$-$C$ bimodules in the sense of A. Ocneanu, we prove that we can construct a finite dimensional non-degenerate commuting square which produces the inclusions $A \subset B$ and $C^{\rm opp}\subset D^{\rm opp}$ by iterated basic constructions.
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(C)2000 Nobuya Sato [e-mail:nobuya@mi.cias.osakafu-u.ac.jp]