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

For a completely regular space $X$, let $C_B(X)$ be the normed algebra of all bounded continuous scalar-valued mappings on $X$ equipped with pointwise addition and multiplication and the supremum norm and let $C_0(X)$ be its subalgebra consisting of mappings vanishing at infinity. For a non-vanishing closed 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 find necessary and sufficient (algebraic) conditions on $H$ such that the spectrum $\mathfrak{sp}(H)$ satisfies (topological) properties such as the Lindel\"{o}f property, $\sigma$-compactness, countable compactness, pseudocompactness and paracompactness.