Estimations of the numerical index of a JB${}^{\ast }$-triple

We prove that every commutative JB${}^{\ast }$-triple has numerical index one. We also revisit the notion of commutativity in JB${}^{\ast }$-triples to show that a JBW${}^{\ast }$-triple $M$ has numerical index one precisely when it is commutative, while $e^{-1}\le n(M)\le 2^{-1}$ otherwise. Consequently, a JB${}^{\ast }$-triple $E$ is commutative if and only if $n(E^{\ast })=1$ (equivalently, $n(E^{\ast \ast })=1$). In the general setting we prove that the numerical index of each JB${}^{\ast }$-triple $E$ admitting a non-commutative element also satisfies $e^{-1}\le n(M)\le 2^{-1}$, and the same holds when the bidual of $E$ contains a Cartan factor of rank $\ge 2$ in its atomic part.