Ternary weak amenability of the bidual of a JB∗-triple

Abstract

Beside the triple product induced by ultrapowers on the bidual of a JB${}^{\ast }$-triple, we assign a triple product to the bidual, $E^{\ast \ast }$, of a JB-triple system $E$, and we show that, under some mild conditions, it makes $E^{\ast \ast }$ a JB-triple system. To study ternary $n$-weak amenability of $E^{\ast \ast }$, 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 $E^{\ast \ast }$ imply the same property for $E?$ In this respect, we show that if the bidual of a JB${}^{\ast }$-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 $n\ge 2$, and the case $n=1$ requires some additional assumptions.