on the theory of compact operators in von neumann ... - Project Euclid
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on the theory of compact operators in von neumann ... - Project Euclid
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PACIFIC JOURNAL OF MATHEMATICS Vol. 79, No. 1, 1978
ON THE THEORY OF COMPACT OPERATORS IN VON NEUMANN ALGEBRAS II VICTOR
KAFTAL
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.
In the second section we shall first re-obtain an equivalent form of Fillmore's theorem and then we shall generalize it to the case of normal operators, thus extending the Berg-Sikonia-Halmos theorem (see [2], [13], and [8]) to infinite factors. Finally we shall examine the possibility of choosing B in the von Neumann algebra generated by A and we shall generalize to normal operators a connected theorem by Zsido' [15]. We wish to thank M. Sonis for having called our attention to this problem. 1* The Weyl-von Neumann theorem in von Neumann factors. Let if be a Hubert space, ^/ be a countably decomposable (i.e., σ finite) semi-finite (i.e., type I or II) von Neumann factor on H, *$/' be its commutant and ^ be the ideal of compact operators of J ^ that is, the norm closure of the ideal of the operators A e Jzf with range projection RA finite relatively to j& (finite operators for short). Let Tr (TrO be a semi-finite faithful normal trace on J ^ + (J^' + ) and D {D') be its restriction to the projections of Jzf (J^') We use the normalization of the relative dimensions D and D' for which D(J) = 1 (JD'(J) = 1) when 0 there is a finite projection P and a finite operator S = S* such that: (1) (I-P)/=0
( 2 ) A — S is reduced by P ( 3 ) (A — S)P is diagonal ( 4 ) \\S\\
