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In their recent works L. Zsido' and P. A, Fillmore have extended WeyΓs version of the classical Weyl-von Neumann theorem to infinite semi-finite countably decomposable von Neumann factors, by proving that for every self-adjoint operator A in the factor there is a diagonal operator B = Σ λnEn such that A — B is compact, the En are one-dimensional projections and {λn} is dense in the essential spectrum of A. In this paper we extend the Weyl-von Neumann theorem in a different way. First we extend the von Neumann version of the theorem to both finite and infinite factors by proving that A — B can be chosen as a Hilbert-Schmidt operator of arbitrarily small norm. We have to drop the condition about the λn or the dimension of the En.

On the theory of compact operators in von Neumann algebras. II