ESSENTIALLY QUASINILPOTENT ELEMENTS WITH RESPECT TO ARBITRARY NORM CLOSED TWO-SIDED IDEALS IN VON NEUMANN ALGEBRAS

In this paper we prove that a part of the Riesz decomposition theory for compact operators holds in maximal generality in the realm of von Neumann algebras. More precisely, if an element x of a von Neumann algebra M is essentially quasinilpotent with respect to an arbitrary norm closed twosided ideal of M, then the supremum (in the projection lattice of M) of the kernel projections of all positive integer powers of 1 — x belongs to the ideal. It seems to be an interesting question, whether the above statement holds in arbitrary AW*-algebras.