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

  • Raymond MortiniEmail author
  • Rudolf Rupp
Article
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Abstract

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 \(C(X(\mathcal {R}), \tau )\).

Keywords

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

Mathematics Subject Classification

Primary 46J05 Secondary 46J10 46J40 

Notes

Acknowledgements

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.

References

  1. 1.
    Allan, G.: Introduction to Banach Spaces and Algebras. Oxford University Press, Oxford (2011)zbMATHGoogle Scholar
  2. 2.
    Alling, N., Campbell, L.A.: Real Banach algebras. II. Math. Z. 125, 79–100 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Corach, G., Suárez, F.D.: Extension problems and stable rank in commutative Banach algebras. Topol. Appl. 21, 1–8 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corach, G., Suárez, F.D.: On the stable range of uniform algebras and \(H^\infty \). Proc. Am. Math. Soc. 98, 607–610 (1986)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Dudley, R.M., Norvaiša, R.: Concrete Functional Calculus. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Furutani, K.: A note on the Arens-Royden theorem for real Banach algebras. TRU Math. 15, 19–22 (1979)MathSciNetGoogle Scholar
  7. 7.
    Gamelin, T.W.: Uniform Algebras. Chelsea, New York (1984)zbMATHGoogle Scholar
  8. 8.
    Kulkarni, S.H., Limaye, B.V.: Real Function Algebras. Marcel Dekker, New York (1992)zbMATHGoogle Scholar
  9. 9.
    Li, Bingren: Real Operator Algebras. World Scientific, New Jersey (2003)CrossRefzbMATHGoogle Scholar
  10. 10.
    Mortini, R., Rupp, R.: A Space Odyssey, Extension Problems and Stable Ranks—Accompanied by Introductory Chapters on Point-Set Topology and Banach Algebras– Textbook, Encylopedic Monograph, ca., pp. 2000 (in preparation) Google Scholar
  11. 11.
    Rickart, C.E.: General Theory of Banach Algebras. Krieger Publishing, Florida (1974)zbMATHGoogle Scholar

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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