Published online by Cambridge University Press: 23 September 2009
Let ℳ be a von Neumann algebra acting on a Hilbert space and let
be a von Neumann subalgebra of ℳ. If
is singular in
for every Hilbert space
,
is said to be completely singular in ℳ. We prove that if
is a singular abelian von Neumann subalgebra or if
is a singular subfactor of a type-II1 factor ℳ, then
is completely singular in ℳ.
is separable, we prove that
is completely singular in ℳ if and only if, for every θ∈Aut(
′) such that θ(X)=X for all X ∈ ℳ′, θ(Y)=Y for all Y∈
′. As the first application, we prove that if ℳ is separable (with separable predual) and
is completely singular in ℳ, then
is completely singular in
for every separable von Neumann algebra
. As the second application, we prove that if
1 is a singular subfactor of a type-II1 factor ℳ1 and
2 is a completely singular von Neumann subalgebra of ℳ2, then
is completely singular in
.