The Banach algebra of continuous bounded functions with separable support

Volume 210 / 2012

M. R. Koushesh Studia Mathematica 210 (2012), 227-237 MSC: Primary 46J10, 46J25, 46E25, 46E15; Secondary 54C35, 54D35, 46H05, 16S60. DOI: 10.4064/sm210-3-3


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 $C_s(X)$.


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

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