A note on compact ideal perturbations in semifinite von Neumann algebras

Abstract:

Let $M$ be a semifinite von Neumann algebra and denote by $J(M)$ the closed two-sided ideal generated by the finite projections in $M$. A subspace $S \subset M$ is called local if it is equal to the ultraweak closure of $S \cap J(M)$. If $M = B(H)$ and $J(M) = K(H)$, Fall, Arveson, and Muhly proved that $S + J(M)$ is closed for every local subspace $S$. In this note we prove that if $M$ is a type ${\text {I}}{{\text {I}}_\infty }$, factor, then there exist local subspaces in $M$ which fail to have closed compact ideal perturbations; thus answering in the negative a question of Kaftal, Larson, and Weiss.