# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## The selfadjoint operators of a von Neumann algebra form a conditionally complete latticeHTML articles powered by AMS MathViewer

by Milton Philip Olson
Proc. Amer. Math. Soc. 28 (1971), 537-544 Request permission

## Abstract:

The bounded resolutions of the identity in a von Neumann algebra can be ordered by $\{ {E_S}(u)\} \preccurlyeq \{ {E_T}(u)\}$ if ${E_S}(u) \geqq {E_T}(u),u \in R$. The selfadjoint operators in the algebra are partially ordered by this relation and are shown to form a conditionally complete lattice. The lattice operations are (essentially) defined by ${E_{V{S_\alpha }}}(u) = \wedge {E_{{S_\alpha }}}(u)$ for all $u$ contained in $R$. This order is called spectral order and agrees with the usual order on commutative subalgebras. For positive operators, $S$ is greater than or equal to $T$ in spectral order if and only if ${S^n}$ is greater than or equal to ${T^n}$ in the usual order for all $n \geqq 1$. Kadison’s well-known counterexample is shown to fail. The operator lattice defined by spectral order differs from a vector lattice in the fact that $S \succcurlyeq T$ does not imply that $S + C \succcurlyeq T + C$.
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Additional Information
• © Copyright 1971 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 28 (1971), 537-544
• MSC: Primary 46.65
• DOI: https://doi.org/10.1090/S0002-9939-1971-0276788-1
• MathSciNet review: 0276788