The Banach algebra of continuous bounded functions with separable support

M. R. Koushesh. "The Banach algebra of continuous bounded functions with separable support." Studia Mathematica 210.3 (2012): 227-237. <http://eudml.org/doc/285647>.

@article{M2012,
abstract = {We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_\{s\}(X)$ 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 $C_\{s\}(X)$. Further, we find the dimension of the algebra $C_\{s\}(X)$.},
author = {M. R. Koushesh},
journal = {Studia Mathematica},
keywords = {Banach algebras; Stone-Čech compactification, separable support; locally separable metrizable space; spectrum; functions vanishing at infinity; functions with compact support},
language = {eng},
number = {3},
pages = {227-237},
title = {The Banach algebra of continuous bounded functions with separable support},
url = {http://eudml.org/doc/285647},
volume = {210},
year = {2012},
}

TY - JOUR
AU - M. R. Koushesh
TI - The Banach algebra of continuous bounded functions with separable support
JO - Studia Mathematica
PY - 2012
VL - 210
IS - 3
SP - 227
EP - 237
AB - We prove a commutative Gelfand-Naimark type theorem, by showing that the set $C_{s}(X)$ 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 $C_{s}(X)$. Further, we find the dimension of the algebra $C_{s}(X)$.
LA - eng
KW - Banach algebras; Stone-Čech compactification, separable support; locally separable metrizable space; spectrum; functions vanishing at infinity; functions with compact support
UR - http://eudml.org/doc/285647
ER -