Generalized Triple Homomorphisms and Derivations

Abstract

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.

The structure of triple homomorphisms onto prime algebras

“…When A is endowed with a conjugate-linear involution * , A is called a Banach * -algebra and in this case, A + becomes a Jordan-Banach * -algebra. Every complex Banach * -algebra A is a Jordan triple via the triple product {x, y, z} = 1 2 (x y * z + zy * x) for x, y, z ∈ A (1 •1) and every complex Jordan-Banach * -algebra A is a Jordan triple via the triple product By a triple homomorphism between Banach * -algebras or Jordan-Banach * -algebras we mean with respect to the triple product as given by (1•1) or (1•2) respectively [10,18,36]. A JB * -triple is a complex Banach space E with a continuous triple product {., ., .}…”

“…Recently, triple homomorphisms on C * -algebras and JB * -triples have been investigated in the literature [2,5,10,[13][14][17][18]. In [26] Kaup proved that a linear bijection between two JB * -triples is an isometry if and only if it is a triple isomorphism.…”

“…It is also well known that every linear * -homomorphism between C * -algebras is continuous [23]. In [5] (or see [18]) Barton The notions of Q s (A), the symmetric Martindale algebra of quotients of A and M(A), the multiplier algebra of A, appear naturally in the study of (anti-)automorphisms of algebras [7,27] and in the extension of C * -algebras [3]. Besides, it is noteworthy to mention that the element u and the range of the map φ in Theorem 1•1 are not necessarily contained in A.…”