On Completely Singular von Neumann Subalgebras

Abstract

Let $\M$ be a von Neumann algebra acting on a Hilbert space $\H$, and $\N$ be a singular von Neumann subalgebra of $\M.$ If $\N\tensor\B(\K)$ is singular in $\M\tensor\B(\K)$ for any Hilbert space $\K$, we say $\N$ is \emph{completely singular} in $\M$. We prove that if $\N$ is a singular abelian von Neumann subalgebra or if $\N$ is a singular subfactor of a type $II_1$ factor $\M$, then $\N$ is completely singular in $\M$. For any type $II_1$ factor $\M$, we construct a singular von Neumann subalgebra $\N$ of $\M$ ($\N\neq \M$) such that $\N\tensor\B(l^2(\mathbb{N}))$ is regular (hence not singular) in $\M\tensor \B(l^2(\mathbb{N}))$. If $\H$ is separable, then $\N$ is completely singular in $\M$ if and only if for any $\theta\in Aut(\N')$ such that $\theta(X)=X$ for all $X\in\M'$, then $\theta(Y)=Y$ for all $Y\in\N'$. As an application of this characterization of completely singularity, we prove that if $\M$ is separable (with separable predual) and $\N$ is completely singular in $\M$, then $\N\tensor\L$ is completely singular in $\M\tensor \L$ for any separable von Neumann algebra $\L$.