Single elements of finite CSL algebras

Abstract:

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