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

## TL;DR

This paper establishes that commutative JB$^*$-triples have a numerical index of one and characterizes the numerical index of JB$^*$-triples based on their commutativity properties, providing new insights into their structure.

## Contribution

It proves that all commutative JB$^*$-triples have numerical index one and characterizes the numerical index for non-commutative JB$^*$-triples, linking it to their structural properties.

## Key findings

- Commutative JB$^*$-triples have numerical index one.
- JB$^*$-triples with non-commutative elements have numerical index between e^{-1} and 1/2.
- The numerical index is one if and only if the triple is commutative.

## 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 $e^{-1}\leq n(M) \leq 2^{-1}$ 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.