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Minimal upper bounds of commuting operators

Authors: Charles Akemann and Nik Weaver
Journal: Proc. Amer. Math. Soc. 124 (1996), 3469-3476
MSC (1991): Primary 47C15, 46L10
MathSciNet review: 1343677
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Abstract: Let $(x_{i})$ be a finite collection of commuting self-adjoint elements of a von Neumann algebra $ \mathcal {M}$. Then within the (abelian) C*-algebra they generate, these elements have a least upper bound $x$. We show that within $ \mathcal {M}$, $x$ is a minimal upper bound in the sense that if $y$ is any self-adjoint element such that $x_{i} \leq y \leq x$ for all $i$, then $y = x$. The corresponding assertion for infinite collections $(x_{i})$ is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if $ \mathcal {N} \subset \mathcal {M}$ are von Neumann algebras, $\Phi : \mathcal {M} \to \mathcal {N}$ is a faithful conditional expectation, and $x \in \mathcal {M}$ is positive, then $\Phi (x^{n})^{1/n}$ converges in the strong operator topology to the ``spectral order majorant'' of $x$ in $ \mathcal {N}$.

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Additional Information

Charles Akemann
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

Nik Weaver
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106

Received by editor(s): June 5, 1995
Additional Notes: This research was partially supported by NSF grant DMS-9424370.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

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