The Inner Corona Algebra of a C ₀(X)-Algebra

Abstract

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 A ^s be the algebra of strong*-continuous functions from X to K(H). Then A ^s /A is the inner corona algebra of A. We show that if X has no isolated points, then A ^s /A is an essential ideal of the corona algebra of A, and Prim(A ^s /A), the primitive ideal space of A ^s /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 A ^s /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 A ^s /A. Several of the results are obtained in the more general setting of C ₀(X)-algebras.