Isomorphisms of certain CSL algebras
type: Item
instance of: scholarly article
title: Isomorphisms of certain CSL algebras
zbMATH Open document ID: 0635.47040
DOI: 10.1016/0022-1236(86)90039-X
author: R. L. Moore
author: Frank L. Gilfeather
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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 compositio…
reviewed by: Marco Degiovanni
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 4034476
zbMATH Keywords: subspace lattice on a complex Hilbert space
zbMATH Keywords: orthogonal projections
zbMATH Keywords: commutative subspace lattice
zbMATH Keywords: lattice of range projections
zbMATH Keywords: algebraic isomorphisms between CSL algebras
zbMATH Keywords: quasi-spatial implementation
zbMATH Keywords: CDC algebra
MaRDI profile type: Publication
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full work available at URL: 0022-1236(86)90039-x
OpenAlex ID: W2037133309
description: scientific article; zbMATH DE number 4034476