Abstract
We are concerned with the development of the more general real case of the classical theorem of Gelfand on representation of a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. To that end, we use only intrinsic methods which do not depend on the complexification of the algebra, and obtain two representation theorems for commutative unital real Banach algebras as algebras of continuous real (respectively, complex) functions on the compact space of real-valued (respectively, complex-valued) \(\mathbb R\)-algebra homomorphisms.
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References
Albiac, F., Briem, E.: Representations of real Banach algebras. J. Aust. Math. Soc. 88, 289–300 (2010)
Albiac, F., Briem, E.: Real Banach algebras as \(\cal{C}({\cal{K}})\) algebras. Q. J. Math. 63(3), 513–524 (2012)
Albiac, F., Kalton, N.J.: A characterization of real \({\cal{C}}(K)\)-spaces. Am. Math. Mon. 114(8), 737–743 (2007)
Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233, 2nd edn. Springer, Cham (2016)
Arens, R.: Representation of *-algebras. Duke Math. J. 14, 269–282 (1947)
Bonsall, F.F., Stirling, D.S.G.: Square roots in Banach \(^{\ast }\)-algebras. Glasg. Math. J. 13, 74 (1972)
Dales, H.G.: Banach Algebras and Automatic Continuity. London Mathematical Society Monographs. New Series, vol. 24. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (2000), p. xviii+907
Kaplansky, I.: Normed algebras. Duke Math. J. 16(399–418), 0012–7094 (1949)
Kulkarni, S.H., Limaye, B.V.: Gel’fand–Naimark theorems for real Banach \(\ast \)-algebras. Math. Jpn. 25(5), 545–558 (1980)
Kulkarni, S.H., Limaye, B.V.: Real Function Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 168, Marcel Dekker Inc., New York, p. viii+186 (1992)
Oliver, H.W.: Noncomplex methods in real Banach algebras. J. Funct. Anal. 6, 401–411 (1970)
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The authors would like to thank the anonymous referees for their helpful suggestions.
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The writing of this article was completed during a visit of the first-named author to the Department of Mathematics at the Science Institute, University of Iceland in Reikiavik, in the Spring of 2019. F. Albiac would like to thank Campus Iberus for the management of the training activities in international mobility for faculty members within the Erasmus+ program of the European Commission, and to express his gratitude to the host Institution, most especially to Prof. Eggert Briem, for his hospitality and generosity during his stay. He also acknowledges the support from the Spanish Research Grants Operators, lattices, and structure of Banach spaces, with reference MTM2016-76808-P, and Análisis Vectorial, Multilineal y Aproximación, with reference PGC2018-095366-B-I00.
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Albiac, F., Briem, E. Gelfand theory for real Banach algebras. RACSAM 114, 163 (2020). https://doi.org/10.1007/s13398-020-00894-4
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DOI: https://doi.org/10.1007/s13398-020-00894-4
Keywords
- Real commutative Banach algebra
- Real algebra homomorphism
- \(\mathcal {C}({\mathcal K})\)-space
- Representation of algebras
- Gelfand theory