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On the ranks of single elements
of reflexive operator algebras


Authors: W. E. Longstaff and Oreste Panaia
Journal: Proc. Amer. Math. Soc. 125 (1997), 2875-2882
MSC (1991): Primary 47C05
DOI: https://doi.org/10.1090/S0002-9939-97-03968-3
MathSciNet review: 1402872
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Abstract: For any completely distributive subspace lattice $\mathfrak {L}$ on a real or complex reflexive Banach space and a positive integer $n$, necessary and sufficient (lattice-theoretic) conditions are given for the existence of a single element of $Alg\mathfrak {L}$ of rank $n$. Similar conditions are given for the existence of single elements of infinite rank. From this follows a relatively simple lattice-theoretic condition which characterises when every non-zero single element has rank one. Slightly stronger results are obtained for the case where $\mathfrak {L}$ is finite, including the fact that every single element must then be of finite rank.


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Additional Information

W. E. Longstaff
Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia
Email: longstaf@maths.uwa.edu.au

Oreste Panaia
Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia
Email: oreste@maths.uwa.edu.au

DOI: https://doi.org/10.1090/S0002-9939-97-03968-3
Received by editor(s): April 1, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society