Single elements of finite CSL algebras
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- by W. E. Longstaff and Oreste Panaia PDF
- Proc. Amer. Math. Soc. 129 (2001), 1021-1029 Request permission
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.References
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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
- Received by editor(s): June 20, 1999
- Published electronically: October 11, 2000
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 1021-1029
- MSC (2000): Primary 47L35; Secondary 47C05
- DOI: https://doi.org/10.1090/S0002-9939-00-05714-2
- MathSciNet review: 1814141