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 | References | Similar Articles | Additional Information

Abstract: An element of an (abstract) algebra is a *single element* of if and imply that or . Let be a real or complex reflexive Banach space, and let be a finite atomic Boolean subspace lattice on , with the property that the vector sum is closed, for every . For any subspace lattice the single elements of Alg are characterised in terms of a coordinatisation of involving . (On separable complex Hilbert space the finite distributive subspace lattices 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:
https://doi.org/10.1090/S0002-9939-00-05714-2

Received by editor(s):
June 20, 1999

Published electronically:
October 11, 2000

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society