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For a completely regular space $X$, let $C_B(X)$ be the normed algebra of allbounded continuous scalar-valued mappings on $X$ equipped with pointwiseaddition and multiplication and the supremum norm and let $C_0(X)$ be itssubalgebra consisting of mappings vanishing at infinity. For a non-vanishingclosed ideal $H$ of $C_B(X)$ we study properties of its spectrum$\mathfrak{sp}(H)$ which may be characterized as the unique locally compact(Hausdorff) space $Y$ such that $H$ and $C_0(Y)$ are isometrically isomorphic.We concentrate on compactness properties of $\mathfrak{sp}(H)$ and findnecessary and sufficient (algebraic) conditions on $H$ such that the spectrum$\mathfrak{sp}(H)$ satisfies (topological) properties such as the Lindel\"{o}fproperty, $\sigma$-compactness, countable compactness, pseudocompactness andparacompactness.

On closed non-vanishing ideals in $C_B(X)$ II; compactness properties