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