Automatic continuity of derivations on \(C^\ast\)-algebras and \(JB^\ast\)-triples
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.
scientific article; zbMATH DE number 6329127
scientific article