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

Abstract

We prove that if an inclusion of von Neumann algebras N ⊂ M has a conditional expectation E : M → N satisfying the finite index condition E ( x) ≥ cx, ∀ x ϵ M +, for some c > 0, then N ⊂ M satisfies the relative version of Dixmier's property on averaging elements by unitaries in N , i.e., for any x ϵ M , the norm closure of the convex hull of { uxu∗ ∥ u unitary element in N } contains elements of N ′ ∩ M . Moreover, in the case N, M are factors of type II 1 and N has separable predual, the finiteness of the index of the inclusion is proved equivalent to the relative Dixmier property and to the property that a normal state on N has only normal state extensions to N . We give applications of these results.