For a completely regular space X, let CB(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 C0(X) be its subalgebra consisting of mappings vanishing at infinity. For a non-vanishing closed ideal H of CB(X) we study properties of its spectrum sp(H) which may be characterized as the unique locally compact (Hausdorff) space Y such that H and C0(Y) are isometrically isomorphic. We concentrate on compactness properties of sp(H) and find necessary and sufficient (algebraic) conditions on H such that the spectrum sp(H) satisfies (topological) properties such as the Lindelöf property, σ-compactness, countable compactness, pseudocompactness and paracompactness.