The relative Dixmier property for inclusions of von Neuman algebras of finite index

We prove that if an inclusion of von Neumann algebras N subset of M has a c onditional expectation epsilon: M --> N satisfying the finite index conditi on epsilon(x) greater than or equal to cx,For All x is an element of M+, fo r some c > 0, then N subset of M satisfies the relative version of Dixmier' s property on averaging elements by unitaries in N, i.e., for any x is an e lement of M, the norm closure of the convex hull of {uxu* \ u unitary eleme nt in N} contains elements of N' boolean AND M. Moreover, in the case N, M are factors of type II1 and N has separable predual, the finiteness of the index of the inclusion is proved equivalent to the relative Dixmier propert y and to the property that a normal stare on N has only normal state extens ions to itt. We give applications of these results. (C) Elsevier, Paris.