Approximation and convex decomposition by extremals and the $\lambda$-function in JBW*-triples

We establish new estimates to compute the $\lambda$-function of Aron and Lohman on the unit ball of a JB${}^{\ast }$-triple. It is established that for every Brown-Pedersen quasi-invertible element $a$ in a JB${}^{\ast }$-triple $E$ we have $$\text{dist}(a,\mathfrak{E}(E_1))=\max \{1-m_q(a),\mathrm{\parallel }a\mathrm{\parallel }-1\},$$ where $\mathfrak{E}(E_1)$ denotes the set of extreme points of the closed unit ball $E_1$ of $E$. It is proved that $\lambda (a) = \frac{1+m_q (a)}{2},$foreveryBrown-Pedersenquasi-invertibleelement$a$in$E_1$,where$m_q (a)$isthesquarerootofthequadraticconormof$a$. For an element $a$ in $E_1$ which is not Brown-Pedersen quasi-invertible we can only estimate that $\lambda (a)\le \frac{1}{2}(1-{\alpha }_q(a))\mathrm{.}$ A complete description of the $\lambda$-function on the closed unit ball of every JBW${}^{\ast }$-triple is also provided, and as a consequence, we prove that every JBW${}^{\ast }$-triple satisfies the uniform $\lambda$-property.