Single elements of finite CSL algebras

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.