Derivations of CDC algebras
type: Item
instance of: scholarly article
title: Derivations of CDC algebras
zbMATH Open document ID: 1105.47030
author: Fangyan Lu
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-11-15T00:00:00Z
review text: Let \(H\) be a Hilbert space and let \({\mathcal A}\) be a subalgebra of \(B(H)\). We say that a derivation \(\delta:{\mathcal A}\to B(H)\) is quasi-spatial if there exists a densely defined, closed linear operator \(T: \text{Dom}(T)\to H\) such that \(\text{Dom}(T)\) is invariant under every operator in \(\mathcal A\) and \(\delta(A)x = (TA -AT)x\) for all \(A\in \mathcal A\) and \(x\in \text{Dom}(T)\). The main result of the present paper characterizes quasi-spatial derivations on \(\mathcal A = \text{Alg}\,\mathcal L\), where \(\mathcal L\) is a completely distributive and commutative lattice on a separable Hilbert space \(H\). It states that a derivation \(\delta:\text{Alg}\,{\mathcal L}\to B(H)\) is quasi-spatial if and only if \(\delta(R)\) maps the kernel of \(R\) into the range of \(R\) for every \(R\) from the algebra generated by all rank one operators in \(\text{Alg}\,\mathcal L\). An analogous result for isomorphisms was obtained earlier by \textit{F.~Gilfeather} and \textit{R.~L.\ Moore} [J.~Funct.\ Anal.\ 67, 264--291 (1986; Zbl 0635.47040)].
reviewed by: Matej Brešar
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46K50
Mathematics Subject Classification ID: 47L35
Mathematics Subject Classification ID: 06D10
zbMATH DE Number: 5072938
zbMATH Keywords: CDC algebra
zbMATH Keywords: derivation
zbMATH Keywords: quasispatiality
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.10.021
OpenAlex ID: W1970300587
cites work: Operator algebras and invariant subspaces
cites work: Derivations of nest algebras
cites work: Q3994569
cites work: Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations
cites work: Isomorphisms of certain CSL algebras
cites work: Local derivations and local automorphisms.
cites work: Q3972049
cites work: Completely distributive latices
cites work: A Note on Rank-One Operators in Reflexive Algebras
cites work: Derivations, local derivations and atomic boolean subspace lattices
cites work: Strongly Reflexive Lattices
cites work: Operators of Rank One in Reflexive Algebras
cites work: Lie derivations of certain CSL algebras
cites work: Bilocal derivations of standard operator algebras
DOI: 10.1016/J.JMAA.2005.10.021
Recommended article: Derivations of CSL algebras
Recommended article: Q3333598
Recommended article: Derivations and Cayley derivations of generalized Cayley-Dickson algebras
Recommended article: Q5299116
Recommended article: Derivations on FCIN algebras
Recommended article: Derivations of commutative algebras
Recommended article: Q3477492
Recommended article: Q4335988
Recommended article: Q4235230
Recommended article: Q5717193
description: scientific article; zbMATH DE number 5072938