Estimations of the Numerical Index of a JB ^* -Triple

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≤ n(M) ≤ 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≤ n(E) ≤ 2^-1 , and the same holds when the bidual of E contains a Cartan factor of rank ≥ 2 in its atomic part.