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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Single elements of finite CSL algebras


Authors: W. E. Longstaff and Oreste Panaia
Journal: Proc. Amer. Math. Soc. 129 (2001), 1021-1029
MSC (2000): Primary 47L35; Secondary 47C05
Published electronically: October 11, 2000
MathSciNet review: 1814141
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Abstract: An element $s$ of an (abstract) algebra ${\mathcal{A}}$ is a single element of ${\mathcal{A}}$ if $asb=0$ and $a,b\in {\mathcal{A}}$imply that $as=0$ or $sb=0$. Let $X$ be a real or complex reflexive Banach space, and let ${\mathcal{B}}$ be a finite atomic Boolean subspace lattice on $X$, with the property that the vector sum $K+L$ is closed, for every $K,L\in {\mathcal{B}}$. For any subspace lattice ${\mathcal{D}}\subseteq {\mathcal{B}}$the single elements of Alg ${\mathcal{D}}$ are characterised in terms of a coordinatisation of ${\mathcal{D}}$ involving ${\mathcal{B}}$. (On separable complex Hilbert space the finite distributive subspace lattices ${\mathcal{D}}$ which arise in this way are precisely those which are similar to finite commutative subspace lattices. Every distributive subspace lattice on complex, finite-dimensional Hilbert space is of this type.) The result uses a characterisation of the single elements of matrix incidence algebras, recently obtained by the authors.


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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: http://dx.doi.org/10.1090/S0002-9939-00-05714-2
PII: S 0002-9939(00)05714-2
Received by editor(s): June 20, 1999
Published electronically: October 11, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society