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