Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations
type: Item
instance of: scholarly article
title: Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations
zbMATH Open document ID: 0564.47021
DOI: 10.1016/0022-1236(84)90009-0
author: Alan Hopenwasser
author: David R. Larson
author: Frank L. Gilfeather
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of \({}^*\)-algebras. Tomita's tensor product commutation theorem can be restated in the form \(Alg({\mathcal L}_ 1\otimes {\mathcal L}_ 2)=Alg {\mathcal L}_ 1\otimes Alg {\mathcal L}_ 2,\) where each \({\mathcal L}_ i\) is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each \({\mathcal L}_ i\) is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if \({\mathcal L}\) is a finite-width commutative subspace lattice and \({\mathcal K}\) is the set of compact operators then the quasitriangular algebra \(Alg {\mathcal L}+{\mathcal K}\) is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 3899614
zbMATH Keywords: Reflexive algebras
zbMATH Keywords: invariant subspace lattice
zbMATH Keywords: Tomita's tensor product commutation theorem
zbMATH Keywords: reflexive ortho-lattice
zbMATH Keywords: tensor product of nest algebras
zbMATH Keywords: Hochschild cohomology
zbMATH Keywords: CSL algebras
zbMATH Keywords: irreducible tridiagonal algebras with finite-width commutative lattices
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90009-0
OpenAlex ID: W2059523225
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description: scientific article; zbMATH DE number 3899614