First, the strong Stone–Weierstrass conjecture for postliminal ${\rm JB}^*$ -triples is proved. This is combined with the recent classification of ${\rm JB}^*$ -triples $A$ for which $\partial_e(A^*_1)$ is ${\rm weak}^*$ dense in $A^*_1$ , and the weak Stone–Weierstrass theorem for ${\rm JB}^*$ -triples is obtained. If $B$ is a ${\rm JB}^*$ -subtriple of a ${\rm JB}^*$ -triple $A$ such that $B$ separates $\overline{\partial_e(A^*_1)}\cup\{0\}$ , then $B=A$ .