On finiteness of the set of intermediate subfactors

Abstract

For type I I 1 II_1 factors N ⊂ L N\subset L with [ L : N ] > ∞ [L:N]>\infty , we show that the sets L 1 = { M ∈ L ( N ⊂ L ) : N ′ ∩ L ⊂ M } \mathcal {L}_1\!=\!\{M\!\in \! \mathcal {L}(N\!\subset \! L)\colon N’\cap L\! \subset \! M\} and L 2 = { M ∈ L ( N ⊂ L ) : N ′ ∩ L = M ′ ∩ L } \mathcal {L}_2\!=\!\{M\!\in \! \mathcal {L}(N\!\subset \! L)\colon N’\cap L \!=\!M’\cap L\} are finite. Moreover, L ( N ⊂ L ) \mathcal {L}(N\subset L) , the set of intermediate subfactors, is finite if and only if it is equal to L 1 ∪ L 2 \mathcal {L}_1\cup \mathcal {L}_2 . If N N is an irreducible subfactor, then we recover a result of Y. Watatani.