The Banach algebra of continuous bounded functions with separable support
M. R. Koushesh
Studia Mathematica, Tome 209 (2012), p. 227-237 / Harvested from The Polish Digital Mathematics Library

We prove a commutative Gelfand-Naimark type theorem, by showing that the set Cs(X) of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable, is moreover non-normal; in addition C₀(Y) = C₀₀(Y). When the underlying field of scalars is the complex numbers, the space Y coincides with the spectrum of the C*-algebra Cs(X). Further, we find the dimension of the algebra Cs(X).

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:285647
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     title = {The Banach algebra of continuous bounded functions with separable support},
     journal = {Studia Mathematica},
     volume = {209},
     year = {2012},
     pages = {227-237},
     zbl = {1275.46036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-3-3}
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M. R. Koushesh. The Banach algebra of continuous bounded functions with separable support. Studia Mathematica, Tome 209 (2012) pp. 227-237. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-3-3/