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 StrÃ¶h

$1$ and LÃ¡szlÃ³ ZsidÃ³

$2$
Author institution:

$1$ Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, REPUBLIC OF SOUTH AFRICA

$2$ Dipartimento di Matematica, UniversitÃ di Roma â€œTor Vergataâ€, 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 â€“ x belongs to the ideal. It seems to be an interesting question, whether the above statement holds in arbitrary AW*-algebras.

Keywords: Essentially quasinilpotent elements, norm closed ideals and von Neumann algebras.

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