The Weyl group and the normalizer of a conditional expectation

Abstract

We define a discrete group W(E) associated to a faithful normal conditional expectation E : M → N for N ⊆ M von Neuman algebras. This group shows the relation between the unitary group U_N and the normalizer N_E of E, which can be also considered as the isotropy of the action of the unitary group U_M of M on E. It is shown that W(E) is finite if dim Z(N) < ∞ and bounded by the index in the factor case. Also sharp bounds of the order of W(E) are founded. W(E) appears as the fibre of a covering space defined on the orbit of E by the natural action of the unitary group of M. W(E) is computed in some basic examples.