Single elements of finite CSL algebras

Abstract

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