Let M be a unital \(\hbox {JB}^*\)-algebra whose closed unit ball is denoted by \({\mathcal {B}}_M\). Let \(\partial _e({\mathcal {B}}_M)\) denote the set of all extreme points of \({\mathcal {B}}_M\). We prove that an element \(u\in \partial _e({\mathcal {B}}_M)\) is a unitary if and only if the set$$\begin{aligned} {\mathcal {M}}_{u} = \{e\in \partial _e({\mathcal {B}}_M) : \Vert u\pm e\Vert \le \sqrt{2} \} \end{aligned}$$contains an isolated point. This is a new geometric characterisation of unitaries in M in terms of the set of extreme points of \({\mathcal {B}}_M\).

中文翻译：

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$$ \ hbox {JB} ^ * $$ JB ∗-代数的Unit的度量表征

令M为单位\（\ hbox {JB} ^ * \）-代数，其闭合单位球由\（{\ mathcal {B}} _ M \）表示。令\（\ partial _e（{\ mathcal {B}} _ M} \）表示\（{\ mathcal {B}} _ M \）的所有极值的集合。我们证明元素\（u \ in \ partial _e（{\ mathcal {B}} _ M）\）是单一的，并且仅当集合$$ \ begin {aligned} {\ mathcal {M}} _ { u} = \ {e \ in \ partial _e（{\ mathcal {B}} _ M）：\ Vert u \ pm e \ Vert \ le \ sqrt {2} \} \ end {aligned} $$包含一个孤立点。根据\（{\ mathcal {B}} _ M \）的极点集，这是M中unit的新几何特征。