Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - October 2009 - Volume 52, Issue 03  

Research Article

On completely singular von Neumann subalgebras

Junsheng Fanga1 p1

a1 Department of Mathematics, University of New Hampshire, Durham, NH 03824, USA; Email: (jfang@cisunix.unh.edu)

Abstract

Let xs2133 be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$ and let $\mathcal{N}$ be a von Neumann subalgebra of xs2133. If $\mathcal{N}\operatorname{\bar{\otimes}}\mathcal{B}(\mathcal{K})$ is singular in $\mathcal{M}\operatorname{\bar{\otimes}}\mathcal{B}(\mathcal{K})$ for every Hilbert space $\mathcal{K}$, $\mathcal{N}$ is said to be completely singular in xs2133. We prove that if $\mathcal{N}$ is a singular abelian von Neumann subalgebra or if $\mathcal{N}$ is a singular subfactor of a type-II1 factor xs2133, then $\mathcal{N}$ is completely singular in xs2133. $\mathcal{H}$ is separable, we prove that $\mathcal{N}$ is completely singular in xs2133 if and only if, for every θxs2208Aut($\mathcal{N}$′) such that θ(X)=X for all X xs2208 xs2133′, θ(Y)=Y for all Yxs2208$\mathcal{N}$′. As the first application, we prove that if xs2133 is separable (with separable predual) and $\mathcal{N}$ is completely singular in xs2133, then $\mathcal{N}\operatorname{\bar{\otimes}}\mathcal{L}$ is completely singular in $\mathcal{M}\operatorname{\bar{\otimes}}\mathcal{L}$ for every separable von Neumann algebra $\mathcal{L}$. As the second application, we prove that if $\mathcal{N}$1 is a singular subfactor of a type-II1 factor xs21331 and $\mathcal{N}$2 is a completely singular von Neumann subalgebra of xs21332, then $\mathcal{N}_1\operatorname{\bar{\otimes}}\mathcal{N}_2$ is completely singular in $\mathcal{M}_1\operatorname{\bar{\otimes}}\mathcal{M}_2$.

(Received August 16 2007)

Keywords

  • singular von Neumann subalgebras;
  • completely singular von Neumann subalgebras;
  • tensor products of von Neumann algebras

2000 Mathematics subject classification

  • Primary 46L10;
  • 46L07

Correspondence:

p1 Present address: Department of Mathematics, Texas A & M University, College Station, TX, 77843-3368, USA (jfang@math.tamu.edu).