Automatic continuity of derivations on \(C^\ast\)-algebras and \(JB^\ast\)-triples
type: Item
instance of: scholarly article
title: Automatic continuity of derivations on \(C^\ast\)-algebras and \(JB^\ast\)-triples
zbMATH Open document ID: 1325.46051
author: Antonio M. Peralta
author: Bernard Russo
published in: Journal of Algebra
publication date: 2014-08-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.0096
reviewed by: Lajos Molnár
Mathematics Subject Classification ID: 46H40
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 46K70
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46H25
zbMATH DE Number: 6329127
zbMATH Keywords: derivation
zbMATH Keywords: Jordan derivation
zbMATH Keywords: triple derivation
zbMATH Keywords: Jordan algebra
zbMATH Keywords: \(C^*\)-algebra
zbMATH Keywords: \(JB^*\)-algebra
zbMATH Keywords: Jordan triple
zbMATH Keywords: \(JB^*\)-triple
zbMATH Keywords: Jordan triple module
zbMATH Keywords: automatic continuity
MaRDI profile type: Publication
OpenAlex ID: W2963856031
arXiv ID: 1208.0096
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DOI: 10.1016/J.JALGEBRA.2013.10.017
review text: The authors study the continuity of derivations from a \(JB^*\)-triple, \(E\), into a so-called Banach (Jordan) triple \(E\)-module. This newly introduced latter class of objects includes, besides Banach modules over Banach algebras and Banach Jordan modules over Banach Jordan algebras, the dual space of any real or complex \(JB^*\)-triple. The basic tool in the study is the quadratic annihilator of a submodule.NEWLINENEWLINEThe authors find precise conditions in order that a derivation from a complex \(JB^*\)-triple, \(E\), into a Banach (Jordan) triple \(E\)-module is continuous. They prove that every derivation from a real or complex \(JB^*\)-triple into itself or into its dual space is automatically continuous. The former statement has been known before, but the authors give a completely different proof here. The latter statement is new and has significance for the recent study of ternary weak amenability.NEWLINENEWLINEA celebrated result of B. E. Johnson states that every bounded Jordan derivation from a \(C^*\)-algebra, \(A\), to a Banach \(A\)-bimodule is an associative derivation. Here the authors remove the continuity assumption from this result. As side-product, they obtain a new proof for the famous theorem due to J. R. Ringrose stating that every derivation from a \(C^*\)-algebra, \(A\), into a Banach \(A\)-bimodule is continuous.
description: scientific article; zbMATH DE number 6329127