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Proceedings of the American Mathematical Society

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

 

 

Completely distributive CSL algebras
with no complements in $\mathcal C_p$


Author: J. A. Erdos
Journal: Proc. Amer. Math. Soc. 124 (1996), 1127-1131
MSC (1991): Primary 47D25; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-96-03134-6
MathSciNet review: 1301023
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Abstract: Anoussis and Katsoulis have obtained a criterion for the space $\operatorname{Alg}\,\mathcal L\cap\mathcal C_p$ to have a closed complement in $\mathcal C_p$, where $\mathcal L$ is a completely distributive commutative subspace lattice. They show that, for a given $\mathcal L$, the set of $p$ for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice $\mathcal L$ for which $\operatorname{Alg}\,\mathcal L\cap\mathcal C_p$ has no complement for any $p\not=2$. However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices $\mathcal L$, $\operatorname{Alg}\,\mathcal L\cap\mathcal C_p$ fails to have a complement for any $p\not=2$.


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Additional Information

J. A. Erdos
Email: J.ERDOS@uk.ac.kcl

DOI: https://doi.org/10.1090/S0002-9939-96-03134-6
Keywords: Commutative subspace lattice, complemented subspace, von Neumann-Schatten class
Received by editor(s): October 3, 1994
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society