| | |
Summary: THE WEYL GROUP AND THE NORMALIZER OF A CONDITIONAL
EXPECTATION
M. ARGERAMI AND D. STOJANOFF
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 UN and the normalizer NE of E, which
can be also considered as the isotropy of the action of the unitary group UM 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.
1 Introduction
Let N M be von Neumann algebras and E : M N a conditional expectation.
Denote by UM the unitary group of M and by NE the group
NE = { u UM : E(uxu
) = uE(x)u
, x M },
called the normalizer of E. The group NE has been already studied, between other authors,
by A. Connes ([6]) and Kosaki ([14]) in relation with crossed product inclusions of algebras.
|
