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.