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

In their recent works L. Zsido' and P. A, Fillmore have extended Weys 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 = n E n such that A -B is compact, the E n are one-dimensional projections and { n } 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 n or the dimension of the E n .

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