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

Abstract

We prove that a Le Page-type inequality is also valid for metrically characterizing those JB${}^{\ast }$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB${}^{\ast }$-triple $E$: $(a)$ $E$ is commutative. $(b)$ There exists ${\hat{I}}^3>0$ satisfying $$\Vert \{a,b,\{x,y,z\}\}\Vert \le {\hat{I}}^3\Vert \{x,y,\{a,b,z\}\}\Vert ,for\; alla,b,x,y,z\in E.$$