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Metric Characterisation of Unitaries in $$\hbox {JB}^*$$ JB ∗ -Algebras
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-06-30 , DOI: 10.1007/s00009-020-01556-w
María Cueto-Avellaneda , Antonio M. Peralta

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:

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

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。
更新日期:2020-06-30
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