# Journal of Operator Theory

Volume 42, Issue 2, Fall 1999 pp. 305-330.

Sous-facteurs intermediaires et groupes quantiques mesures**Authors**: Michel Enock

**Author institution:**Institut de Mathematiques de Jussieu, UMR 7586, Case 191, Universite Pierre et Marie Curie, F-75252 Paris Cedex 05, France

**Summary:**Let $M_0\subset M_1$ an irreducible depth 2 inclusion of factors with a faithful semi-finite normal operator-valued weight, verifying a regularity condition, and let $(M_i)_{i\in{\bbb N}}$ the canonical tower; it has been prove d ([8], [7]) that both relative commutants $M_2\cap M^\prime_0$ and $M_3\cap M^\pr ime_1$ bear Woronowicz algebra structures, dual to each other. We show that to every intermediate subfactor $M_0\subset N_0\subset M_1$ can be associated, in a bijective way, a left co-ideal of $M_3\cap M^\prime_1$; this application preserves the lat tice structures on these sets, and we recover and generalize in this way the results of [12], obtained in the case of compact grou ps and compact type Kac algebras by using very different considerations.

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