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Generalised Triple Homomorphisms and Derivations |
26 pages
Published:2013-02-06
- Jorge J. Garcés,
- Antonio M. Peralta,
Abstract
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) |