A revisit of the Arens-Royden and Shilov idempotent theorems for real Banach algebras

  • Raymond MortiniEmail author
  • Rudolf Rupp


Based on the complex case, we present for commutative real Banach algebras \(\mathcal {R}\) several non-algebraic versions of the Arens-Royden theorem and the Shilov idempotent theorem. It will be shown that the Gelfand transform induces a group isomorphism of \(\mathcal {R}^{-1}/\exp \mathcal {R}\) onto \(C(X(\mathcal {R}), \tau )^{-1}/ \exp C(X(\mathcal {R}), \tau )\), where \(C(X(\mathcal {R}),\tau )\) is the algebra of \(\tau \)-symmetric complex-valued functions on the character space \(X(\mathcal {R} )\) of \(\mathcal {R}\) for some specific involution \(\tau \). We will also prove that for any \(\tau \)-symmetric closed-open subset of \(X(\mathcal {R})\) there is an idempotent e in \(\mathcal {R}\) whose Gelfand transform coincides with the characteristic function of E. We apply the real Arens-Royden theorem to show that the Bézout equation \(uf+vg=1\) has a solution in \(\mathcal {R}\) with u invertible if (and only if) it has such a solution in \(C(X(\mathcal {R}), \tau )\).


Real Banach algebras Complexifications and real-izations Group of invertible elements \(\tau \)-Symmetric Idempotents 

Mathematics Subject Classification

Primary 46J05 Secondary 46J10 46J40 



Le premier auteur remercie l’UFR MIM (Mathématiques-Informatique-Mécanique) de l’université de Lorraine, Campus Metz, pour lui avoir donné la possibilité, via une mise à disposition, d’enseigner à l’Université du Luxembourg de 2017 à 2020. Un merci aussi à l’Université du Luxembourg pour avoir soutenu cette démarche.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Département de Mathématiques et Institut Élie Cartan de LorraineUniversité de Lorraine, UMR 7502MetzFrance
  2. 2.Fakultät für Angewandte Mathematik, Physik und AllgemeinwissenschaftenTechnische Hochschule Nürnberg, Georg Simon OhmNürnbergGermany

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