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Journal of Operator Theory

Volume 38, Issue 2, Fall 1997  pp. 379-389.

Essentially quasinilpotent elements with respect to arbitrary norm closed two-sided ideals in von Neumann algebras

Authors Anton Stroh (1) and Laszlo Zsido (2)
Author institution: (1) Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, REPUBLIC OF SOUTH AFRICA
(2) Dipartimento di Matematica, Universita di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, ITALY

Summary:  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 two-sided 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.

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