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It is established that for every Brown-Pedersen quasi-invertible element $a$ in a JB$^*$-triple $E$ we have $$\\hbox{dist} (a, \\mathfrak{E} (E_1)) = \\max \\left\\{ 1- m_q (a) , \\|a\\|-1\\right\\},$$ 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 $a$. 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Siddiqui, Antonio M. Peralta, Fatmah B. Jamjoom, Haifa M. Tahlawi","submitted_at":"2014-04-30T05:47:47Z","abstract_excerpt":"We establish new estimates to compute the $\\lambda$-function of Aron and Lohman on the unit ball of a JB$^*$-triple. It is established that for every Brown-Pedersen quasi-invertible element $a$ in a JB$^*$-triple $E$ we have $$\\hbox{dist} (a, \\mathfrak{E} (E_1)) = \\max \\left\\{ 1- m_q (a) , \\|a\\|-1\\right\\},$$ 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 $a$. 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