Gelfand theory for real Banach algebras
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Gelfand theory for real Banach algebras F. Albiac1
· E. Briem2
Received: 1 June 2019 / Accepted: 30 June 2020 © The Royal Academy of Sciences, Madrid 2020
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) R-algebra homomorphisms. Keywords Real commutative Banach algebra · Real algebra homomorphism · C (K)-space · Representation of algebras · Gelfand theory Mathematics Subject Classification 46J10
1 Introduction In the classical literature, complex Banach algebras have received far greater attention than real Banach algebras perhaps due to the possibility of using the power of analytic function theory via the Gelfand transformation. Scalar multiplication in a complex algebra A is a map
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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F. Albiac [email protected] E. Briem [email protected]
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Math Department-InaMat2, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain
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Science Institute, University of Iceland, 107 Reykjavík, Iceland 0123456789().: V,-vol
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F. Albiac, E. Briem
C × A → A. By considering the restriction of this map to R × A, A can be viewed as a real algebra. Hence every complex Banach algebra is also a real Banach algebra. Thus it is natural to ask what can be said about this larger class. This article intends to contribute to the theory of real Banach algebras by providing new proofs of two known results on the representation (via the Gelfand transform) of a real commutative unital Banach algebra A as a subalgebra of CC (K) (without any additional condition) or as a subalgebra of CR (K) for some compact Hausdorff space K, under an easily checked condition involving the spectral radius. Our work can be seen as a supplement to the monograph on real function algebras by Kulkarni and Limaye [10], to which we refer f
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