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 ∈ 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 D ⊆ 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. Introduction and preliminaries Throughout, X will denote a non-zero real or complex reflexive Banach space, and H will denote a non-zero complex separable Hilbert space. Also, F will denote the real or complex field. By a subspace of X we mean a norm-closed linear manifold and by an operator on X we mean a bounded linear transformation acting on X . The set of operators on X is denoted by B(X). By a subspace lattice on X we mean a family L of subspaces of X satisfying (i) (0), X ∈ L and (ii) for every family {Lγ}Γ of elements of L, ∩ΓLγ ∈ L,∨ΓLγ ∈ L (where ‘∨’ denotes ‘closed linear span’). A subspace lattice on a Hilbert space is commutative if the (orthogonal) projections onto any two of its members commute. (As usual PL will denote the (orthogonal) projection onto the subspace L.) The abbreviation ‘CSL’ will be used for ‘commutative subspace lattice’. A subspace lattice on X is Boolean if it is complemented and distributive, and atomic if each of its elements is the (closed linear) span of the atoms that it contains. The abbreviation ‘ABSL’ will be used for ‘atomic Boolean subspace lattice’. In any subspace lattice L the ‘minus operation’ is the self-map defined by L− = ∨ {M ∈ L : L 6⊆M}, for any element L ∈ L (so that L 6⊆ M ⇒ M ⊆ L−). The annihilator S⊥ of a subset S ⊆ X is, as usual, given by S⊥ = {x∗ ∈ X∗ : x∗(x) = 0, for every x ∈ S}, where X∗ denotes the topological dual of X . For any vectors f ∈ X, e∗ ∈ X∗ the Received by the editors June 20, 1999. 2000 Mathematics Subject Classification. Primary 47L35; Secondary 47C05. c ©2000 American Mathematical Society