Let A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let AS be the algebra of strong*-continuous functions from X to K(H). Then AS/A is the inner corona algebra of A. We show that if X has no isolated points, then AS/A is an essential ideal of the corona algebra of A, and Prim(AS/A), the primitive ideal space of AS/A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of AS/A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of AS/A. Several of the results are obtained in the more general setting of C0(X)-algebras.
- inner corona