A NOTE ON COMPACT IDEAL PERTURBATIONS IN SEMIFINITE VON-NEUMANN-ALGEBRAS

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-of M is called local if it is equal to the ultraweak closure of S and 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 II(infinity) 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.