Isomorphisms of certain CSL algebras
A subspace lattice on a complex Hilbert space \({\mathcal H}\) is a strongly closed lattice \({\mathcal L}\) of orthogonal projections on \({\mathcal H}\) with 0,I\(\in {\mathcal L}\). Given such a lattice \({\mathcal L}\), let Alg \({\mathcal L}\) denote the algebra of all bounded linear operators on \({\mathcal H}\) which leave invariant every element of \({\mathcal L}\). If \({\mathcal L}\) is commutative, Alg \({\mathcal L}\) is called a CSL (commutative subspace lattice) algebra. In this case, \({\mathcal L}\) equals the lattice of all projections invariant for each operator in Alg \({\mathcal L}\) and so Alg \({\mathcal L}\) determines \({\mathcal L}\). (The standard terminology is that \({\mathcal L}\) is a reflexive lattice.) If \({\mathcal L}\) is both commutative and completely distributive (a property which can be defined lattice theoretically or by an equivalent spatial property), then Alg \({\mathcal L}\) is called a CDC algebra. Let \({\mathcal L}_ 1\), \({\mathcal L}_ 2\) be subspace lattices on \({\mathcal H}_ 1\), \({\mathcal H}_ 2\), respectively, and let \(\rho\) :Alg \({\mathcal L}_ 1\to Alg {\mathcal L}_ 2\) be an algebraic isomorphism. If there is an invertible bounded operator T of \({\mathcal H}_ 1\) onto \({\mathcal H}_ 2\) such that \({\mathcal L}_ 2\) equals the lattice of range projections \(\{ran(TET^{- 1}):E\in {\mathcal L}_ 1\}\) and \(\rho (A)=TAT^{-1}\) for A in Alg \({\mathcal L}_ 1\), then \(\rho\) is said to be spatially implemented, while \(\rho\) is quasi-spatially implemented if T is only assumed to be one-one with dense range \({\mathcal D}\) invariant for Alg \({\mathcal L}_ 1\) and \(\rho (A)Tf=TAf\) (A\(\in Alg {\mathcal L}_ 1,f\in {\mathcal D}).\) The aim of the present paper is to establish several properties of algebraic isomorphisms between CSL algebras and to study conditions which ensure spatial or quasi-spatial implementation. Firstly, the authors observe that an isomorphism between two CSL algebras is the composition of a spatially implemented isomorphism and an automorphism; furthermore, the automorphism may be chosen to have some special properties. This result is, in essence due to J. R. Ringrose, who formulated it for nest algebras. Applying this result, together with techniques of E. Christensen, who studied derivations on CSL algebras, the authors show that every algebraic isomorphism between CSL algebras is automatically uniformly bicontinuous. The main result of the paper is then proved. This asserts that, if \(\rho\) :Alg \({\mathcal L}_ 1\to Alg {\mathcal L}_ 2\) is an algebraic isomorphism and Alg \({\mathcal L}_ 1\) is assumed to be a CDC algebra, then \(\rho\) is quasi-spatially implemented by a closed injective linear mapping with dense domain and dense range if and only if \(\rho\) preserves the rank of every finite rank operator. Examples are given to show that some ranks may be preserved by isomorphisms while others are not, and that isomorphisms may be quasi- spatial without being spatial. Other conditions for quasi-spatial or spatiaonditions.
subspace lattice on a complex Hilbert space
algebraic isomorphisms between CSL algebras
scientific article; zbMATH DE number 4034476
scientific article