Estimations of the numerical index of a JB$^*$-triple

Abstract

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