Let A be a semifinite and properly infinite von Neumann algebra of operators of a complex Hilbert space H. Call C e A compact (relative to A) if it is the limit in the norm of elements with finite support. The investigation of analytic properties of compact elements of A, generalizing welt-known classical theorems on compact operators, was started in [2] and is continued in the present paper. Let N i (resp. Ri) be the null (resp. range) projection of(1 C ) ~ where i is a natural number. Contrary to the classical case, the sequences N1 =R2 > R3 ~ "" need not be stationary, even if A is of type I. The question of the finiteness (resp. cofiniteness) of the projection Noo = sup Ni (resp. R~ = infRi) was left open in [2]. In the present paper it is shown that No~ is in fact always finite (and R~ is cofinite). It was shown already in [2] that if N® is finite, then inf(N®, Ro0) = 0 and sup(No~,R~)= 1. This is a decomposition of H in the sense of lattice theory or perhaps continuous geometry. In the present paper this result is refined by introducing the concepts of essentially closed subspaces and essentially topological direct sums. It is shown that the algebraic sum No~(H)+R®(H) is essentially topological direct and essentially closed. The operator 1 C decomposes into the sum of two closed operators affiliated with A, one summand being essentially nilpotent on Noo(H), the other one essentially regular on Roo(H). Throughout this paper H denotes a complex Hilbert space and A a properly infinite semifinite von Neumann algebra of linear operators of H. § 1. Preliminaries