When we have a non-degenerate commuting square of finite
dimensional $C^*$-algebras, we can construct a subfactor in two
ways. One is by a repetition of basic constructions in a
horizontal direction and the other in a vertical direction. We
prove that if one of the two is of finite depth, so is the other.
Furthermore, we prove the two have the same global indices in the
sense of A. Ocneanu. This gives an answer to a question V. F. R.
Jones raised in his talk at Aarhus in June, 1995. We actually
prove a more general result on flatness and also give an example
of a new finite principal graph as its application.
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(C)2000 Nobuya Sato [e-mail:nobuya@mi.cias.osakafu-u.ac.jp]