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Single Elements of Finite CSL Algebras

W. E. Longstaff and Oreste Panaia
Proceedings of the American Mathematical Society
Vol. 129, No. 4 (Apr., 2001), pp. 1021-1029
Stable URL: http://www.jstor.org/stable/2668905
Page Count: 9
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Single Elements of Finite CSL Algebras
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Abstract

An element s of an (abstract) algebra A is a single element of A if asb = 0 and a, b ∈ A imply that as = 0 or sb = 0. Let X be a real or complex reflexive Banach space, and let 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 ∈ B. For any subspace lattice $\mathcal{D} \subseteq \mathcal{B}$ the single elements of Alg D are characterised in terms of a coordinatisation of D involving B. (On separable complex Hilbert space the finite distributive subspace lattices 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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