Ternary weak amenability of the bidual of a JB $^{*}$ -triple

Abstract

Beside the triple product induced by ultrapowers on the bidual of a JB * -triple, we assign a triple product to the bidual, E * * , of a JB-triple system E, and we show that, under some mild conditions, it makes E * * a JB-triple system. To study ternary n-weak amenability of E * * , we need to improve the module structures in the category of JB-triple systems and their iterated duals, which lead us to introduce a new type of ternary module. We then focus on the main question: when does ternary n-weak amenability of E * * imply the same property for E? In this respect, we show that if the bidual of a JB * -triple E is ternary n-weakly amenable, then E is ternary n-quasiweakly amenable. However, for a general JB-triple system, the results are slightly different for n = 1 and n ≥ 2, and the case n = 1 requires some additional assumptions.

Aron–Berner extensions of triple maps with application to the bidual of Jordan Banach triple systems

“…Then Theorem 3.5 implies that (E (2n) , π [n] ) is a again a JB * -triple for each n ∈ N and the triple product π [n] is separately w * -continuous. Niazi, Miri, and the second named author [30] used the extension π [n] and some related module operations for investigating the n-weak amenability of the bidual of a JB * -triple.…”

Local triple derivations from C$^*$-algebras into their iterated duals

“…Trying to endow the dual of a ternary module with ternary module structure, in [13] the authors of this paper improved the previous notion of ternary modules over Jordan triples by introducing a new type of ternary modules and called it ternary module of type (II). We recall both of them in the following:…”

“…We restate the following proposition from [13] which provides a clear picture of the module actions of the iterated duals of a Jordan Banach triple E . …”

“…While the proposed structure make it possible to consider the dual of a Jordan triple as a ternary module, it fails to induce a ternary module structure on the iterated duals of a Jordan triple. To remedy this pathology the authors of this paper proposed another type of ternary module structure in [13], which combined with the previous one exhibit a complete picture of the module structures in the category of Jordan triples (cf. Definition 2.1 and Theorem 2.3 in [13]).…”

“…To remedy this pathology the authors of this paper proposed another type of ternary module structure in [13], which combined with the previous one exhibit a complete picture of the module structures in the category of Jordan triples (cf. Definition 2.1 and Theorem 2.3 in [13]). …”