### Abstract

An element s of an (abstract) algebra A is a single element of A if asb = 0 and a, b is an element of 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 is an element of B. For any subspace lattice D subset of or equal to 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.

Original language | English |
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Pages (from-to) | 1021-1029 |

Journal | Proceedings of the American Mathematical Society |

Volume | 129 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2000 |

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## Cite this

Longstaff, W., & Panaia, O. (2000). Single elements of finite CSL algebras.

*Proceedings of the American Mathematical Society*,*129*(4), 1021-1029. https://doi.org/10.1090/S0002-9939-00-05714-2