Metric Characterisation of Unitaries in \(\hbox {JB}^*\)-Algebras

Abstract

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 \({\mathcal {B}}_M\).

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Acknowledgements

Authors partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European Regional Development Fund Project no. PGC2018-093332-B-I00, Programa Operativo FEDER 2014-2020 and Consejería de Economía y Conocimiento de la Junta de Andalucía Grant number A-FQM-242-UGR18, and Junta de Andalucía grant FQM375. First author also supported by Universidad de Granada, Junta de Andalucía and Fondo Social Europeo de la Unión Europea (Iniciativa de Empleo Juvenil), grant number 6087.

The authors thank the anonymous peer reviewers for their valuable comments.

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Correspondence to Antonio M. Peralta.

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Cueto-Avellaneda, M., Peralta, A.M. Metric Characterisation of Unitaries in \(\hbox {JB}^*\)-Algebras. Mediterr. J. Math. 17, 124 (2020). https://doi.org/10.1007/s00009-020-01556-w

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Keywords

  • Unitaries
  • Geometric unitaries
  • Vertex
  • Extreme points
  • \(\hbox {C}^*\)-algebra
  • \(\hbox {JB}^*\)-algebra
  • \(\hbox {JB}^*\)-triple

Mathematics Subject Classification

  • Primary 46L05
  • 46H70
  • 46L70
  • Secondary 46B20
  • 46K70
  • 46L70
  • 17C65
  • 47C15