Algebraic Isomorphisms and Finite Distributive Subspace Lattices

Let L₁ and L₂ be finite distributive subspace lattices on realor complex Banach spaces. It is shown that every rank-preservingalgebraic isomorphism of AlgL₁ onto AlgL₂ is quasi-spatiallyinduced. If the algebraic isomorphism in question is known onlyto preserve the rank of rank one operators, then it inducesa lattice isomorphism between L₁ and L₂.     

On the ranks of single elements of reflexive operator algebras

For any completely distributive subspace lattice on a real or complex reflexive Banach space and a positive integer , necessary and sufficient (lattice-theoretic) conditions are given for the existence of a single element of of rank . Similar conditions are given for the existence of single elements of infinite rank. From this follows a relatively simple lattice-theoretic condition which characterises when every non-zero single element has rank one. Slightly stronger results are obtained for the case where is finite, including the fact that every single element must then be of finite rank.

     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

Single elements of finite CSL algebras

An element of an (abstract) algebra is a single element of if and imply that or . Let be a real or complex reflexive Banach space, and let be a finite atomic Boolean subspace lattice on , with the property that the vector sum is closed, for every . For any subspace lattice the single elements of Alg are characterised in terms of a coordinatisation of involving . (On separable complex Hilbert space the finite distributive subspace lattices 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.