A Le Page--Kaplansky theorem characterizing commutative JB*-triples

TL;DR

This paper establishes a characterization of commutative JB*-triples using a Le Page-type inequality, providing a new metric criterion for commutativity.

Contribution

It proves that a specific inequality involving triple products characterizes when a JB*-triple is commutative, extending Le Page's approach.

Findings

The inequality holds if and only if the JB*-triple is commutative.

The result provides a metric-based criterion for commutativity.

It generalizes previous characterizations of JB*-triples.

Abstract

We prove that a Le Page-type inequality is also valid for metrically characterizing those JB$^*$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB$^*$-triple $E$: $(a)$ $E$ is commutative. $(b)$ There exists $\gamma>0$ satisfying $$\big\|\{a,b,\{x,y,z\}\}\big\|\leq \gamma \ \! \big\|\{x,y,\{a,b,z\}\}\big\|, \hbox{ for all } a,b,x,y,z\in E.$$