ON COMPLETELY SINGULAR VON NEUMANN SUBALGEBRAS

Let M be a von Neumann algebra acting on a Hilbert space H and let N be a von Neumann subalgebra of M. If N (circle times) over bar) B(K) is singular in M (circle times) over bar B(K) for every Hilbert space K, N is said to be completely singular in M. We prove that if N is a singular abelian von Neumann subalgebra or if M is a singular subfactor of a type-II(1) factor M, then N is completely singular in M. If H is separable, we prove that N is completely singular in M if and only if, for every theta is an element of Aut(N') such that theta(X) = X for all X is an element of M', theta(Y) = Y for all Y is an element of N'. As the first application, we prove that if M is separable (with separable predual) and N is completely singular in M, then N (circle times) over bar L is completely singular in M (circle times) over bar L for every separable von Neumann algebra L. As the second application, we prove that if N(1) is a singular subfactor of a type-II(1) factor M(1) and N(2) is a completely singular von Neumann subalgebra of algebra of M(2), then N(1) (circle times) over bar N(2) is completely singular in M(1) (circle times) over bar M(2).