Let A be a stable, σ-unital C*-algebra which is a C0(X)-algebra, that is, for which there is a continuous map ϕ from Prim(A), the primitive ideal space of A with the hull-kernel topology, to the locally compact Hausdorff space X. We show that there is an injective map L from the lattice of z-ideals of the ring of continuous functions on the completely regular space Im(ϕ) to the lattice of closed ideals of M(A), the multiplier algebra of A. For any two ideals in the range of L, there is a maximal ideal of M(A) containing one but not the other. If Im(ϕ) is infinite, then the corona algebra M(A)/A has at least 2c maximal ideals.
Archbold, R. J., & Somerset, D. W. B. (2012). Ideals in the multiplier and corona algebras of a Co(X)-algebra. Journal of the London Mathematical Society, 85(2), 365-381. https://doi.org/10.1112/jlms/jdr053