Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles
Canadian journal of mathematics, Tome 44 (1992) no. 2, pp. 342-356
Voir la notice de l'article provenant de la source Cambridge University Press
We are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space.In § 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted M, and the set of unital (real) algebra homomorphisms from B into C, denoted by ΦB. This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex.Finally, in § 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras “almost complex C*- algebras” and provide a natural classification scheme.
Pfaffenberger, W. E.; Phillips, J. Commutative Gelfand Theory for Real Banach Algebras: Representations as Sections of Bundles. Canadian journal of mathematics, Tome 44 (1992) no. 2, pp. 342-356. doi: 10.4153/CJM-1992-023-4
@article{10_4153_CJM_1992_023_4,
author = {Pfaffenberger, W. E. and Phillips, J.},
title = {Commutative {Gelfand} {Theory} for {Real} {Banach} {Algebras:} {Representations} as {Sections} of {Bundles}},
journal = {Canadian journal of mathematics},
pages = {342--356},
year = {1992},
volume = {44},
number = {2},
doi = {10.4153/CJM-1992-023-4},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1992-023-4/}
}
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