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Generalised triple homomorphisms and Derivations

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 Jarosz and 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 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 $E^*$ is automatically continuous.

Generalised triple homomorphisms and Derivations

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 Jarosz and 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 Johnson. We shall also consider generalised triple derivations from a Jordan Banach triple into a Jordan Banach triple -module, proving that every generalised triple derivation from a JB-triple into is automatically continuous.
Paper Structure (7 sections, 18 theorems, 133 equations)

This paper contains 7 sections, 18 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The separating space of a generalised triple homomorphism
  4. Automatic continuity
  5. Generalised triple homomorphisms between Jordan Banach triples
  6. Generalised triple derivations from a JB$^*$-triple
  7. Generalised triple derivations whose domain is a C$^*$-algebra

Key Result

Proposition 1

Let $A,B$ be Banach algebras. Every generalised homomorphism $T:A \to B$ is a generalised triple homomorphism when $A$ and $B$ are equipped with the elemental triple product $2\{a,b,c\}=abc+cba$. When $A$ and $B$ are Banach $^*$-algebras and $T$ is a generalised $^*$-homomorphism then $T$ is a gener

Theorems & Definitions (35)

  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Example 4
  • Definition 5
  • Theorem 6
  • proof
  • ...and 25 more