The Banach algebra of continuous bounded functions with separable support

M. R. Koushesh¹

¹ Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156–83111, Iran and School of Mathematics Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395–5746 Tehran, Iran

Studia Mathematica, Tome 210 (2012) no. 3, pp. 227-237

Résumé

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 $C_0(Y)$ 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 $C_0(Y)=C_{00}(Y)$. When the underlying field of scalars is the complex numbers, the space $Y$ coincides with the spectrum of the $ \hbox {C}^*$-algebra $C_s(X)$. Further, we find the dimension of the algebra $C_s(X)$.

Détail

DOI : 10.4064/sm210-3-3

Keywords: prove commutative gelfand naimark type theorem showing set continuous bounded real complex valued functions separable support locally separable metrizable space provided supremum norm banach algebra isometrically isomorphic unique homeomorphism locally compact hausdorff space space which explicitly construct subspace stone ech compactification countably compact non separable moreover non normal addition underlying field scalars complex numbers space coincides spectrum hbox * algebra further dimension algebra