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On completely singular von Neumann subalgebras
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 52 / Issue 3 / October 2009
- Published online by Cambridge University Press:
- 23 September 2009, pp. 607-618
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- Article
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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 
.
and let 
be a von Neumann subalgebra of ℳ. If 
is singular in 
for every Hilbert space 
, 
is completely singular in 
for every separable von Neumann algebra 
. As the second application, we prove that if 
is completely singular in 
.