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

Abstract

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 $$dist(a,\mathfrak{E}(E_1))=\max \left\{1-m_q(a),\Vert a\Vert -1\right\},$$ where $\mathfrak{E}(E_1)$ denotes the set of extreme points of the closed unit ball $E_1$ of $E$. It is proved that $\lambda (a)=\frac{1+m_q(a)}{2},$ for every Brown-Pedersen quasi-invertible element $a$ in $E_1$, where $m_q(a)$ is the square root of the quadratic conorm of $a$. For an element $a$ in $E_1$ which is not Brown-Pedersen quasi-invertible we can only estimate that $\lambda (a)\le \frac{1}{2}(1-{\alpha }_q(a)).$ 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.