Generalised triple homomorphisms and derivations (Q2841819)

This paper presents a contribution to the study of automatic continuity of mappings between complex or real Banach spaces endowed with an algebraic structure. The classical example of this property is that every *-homomorphism from a \(C^*\)-algebra \(A\) into a \(C^*\)-algebra \(B\) is automatically norm-continuous, and, in fact, norm non-increasing. The question of when similar results could be proved for linear mappings between \(C^*\)-algebras led Johnson and Jarosz to study generalized homomorphisms between Banach algebras. The basis of the study is the introduction of a separating space which is designed to measure how much a linear mapping between algebras avoids being an algebraic homomorphism.NEWLINENEWLINE In this paper, the authors study generalized triple homomorphisms between both real and complex Jordan-Banach triples and from a Jordan-Banach triple \(A\) into a Jordan-Banach triple \(A\)-module. The techniques needed to complete this study include a sophisticated investigation into the triple analogue of a separating space and are almost disjoint from those used in working with algebras. As a consequence of the results in this paper, several new contributions to the theory of automatic continuity in algebras, rather than triples, appear.NEWLINENEWLINE The paper contains a plethora of interesting results, far too many to be listed in a review, and, of course, the sharper of these occur when the Jordan-Banach triples are in fact the more natural JB\(^*\)-triples.