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Ideals in $C_B(X)$ arising from ideals in $X$

Volume 245 / 2019

M. R. Koushesh Studia Mathematica 245 (2019), 33-99 MSC: Primary 16S60; Secondary 03E15, 46E15, 46E25, 46H05, 46J10, 46J25, 54C30, 54C35, 54C40, 54D35, 54D60, 54D65. DOI: 10.4064/sm170807-27-9 Published online: 2 July 2018


Let $X$ be a completely regular topological space. We assign to each (set-theoretic) ideal of $X$ an (algebraic) ideal of $C_B(X)$, the normed algebra of continuous bounded scalar-valued mappings on $X$ equipped with the supremum norm. We then prove several representation theorems for those ideals. This is done by associating a certain subspace of the Stone–Čech compactification $\beta X$ to each ideal of $X$. This subspace has a simple representation, and when the ideal is closed, coincides with its spectrum as a Banach algebra. This in particular provides information about the spectrum of those closed ideals of $C_B(X)$ which have such representations. This includes non-vanishing closed ideals of $C_B(X)$ whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of $C_B(X)$ by relating it to the topological properties of their spectrums. This is illustrated by various examples. Our approach is rather topological and makes use of the theory of Stone–Čech compactification.


  • 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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