The Weyl group and the normalizer of a conditional expectation

We define a discrete groupW(E) associated to a faithful normal conditional expectationE : M N forN M von Neuman algebras. This group shows the relation between the unitary groupU _N and the normalizerN _E ofE, which can be also considered as the isotropy of the action of the unitary groupU _M ofM onE. It is shown thatW(E) is finite if dimZ(N)< and bounded by the index in the factor case. Also sharp bounds of the order ofW(E) are founded.W(E) appears as the fibre of a covering space defined on the orbit ofE by the natural action of the unitary group ofM. W(E) is computed in some basic examples.