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# Generalised Triple Homomorphisms and Derivations

Published:2013-02-06
Printed: Aug 2013
• Jorge J. Garcés,
We introduce generalised triple homomorphism between Jordan Banach triple systems as a concept which extends the notion of generalised homomorphism between Banach algebras given by K. Jarosz and B.E. Johnson in 1985 and 1987, respectively. We prove that every generalised triple homomorphism between JB$^*$-triples is automatically continuous. When particularised to C$^*$-algebras, we rediscover one of the main theorems established by B.E. Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple $E$ into a Jordan Banach triple $E$-module, proving that every generalised triple derivation from a JB$^*$-triple $E$ into itself or into $E^*$ is automatically continuous.
 Keywords: generalised homomorphism, generalised triple homomorphism, generalised triple derivation, Banach algebra, Jordan Banach triple, C$^*$-algebra, JB$^*$-triple
 MSC Classifications: 46L05 - General theory of $C^*$-algebras 46L70 - Nonassociative selfadjoint operator algebras [See also 46H70, 46K70] 47B48 - Operators on Banach algebras 17C65 - Jordan structures on Banach spaces and algebras [See also 46H70, 46L70] 46K70 - Nonassociative topological algebras with an involution [See also 46H70, 46L70] 46L40 - Automorphisms 47B47 - Commutators, derivations, elementary operators, etc. 47B49 - Transformers, preservers (operators on spaces of operators)