We prove a commutative Gelfand-Naimark type theorem, by showing that the set 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 . Further, we find the dimension of the algebra .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm210-3-3,
author = {M. R. Koushesh},
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}
}
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/