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 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.
| Original language | English |
|---|---|
| Pages (from-to) | 299-318 |
| Number of pages | 20 |
| Journal | Proceedings of the Edinburgh Mathematical Society |
| Volume | 60 |
| Issue number | 2 |
| Early online date | 19 Sept 2016 |
| DOIs | |
| Publication status | Published - May 2017 |
Bibliographical note
We are grateful to the referee for a number of helpful comments.Keywords
- C0(X)-algebra
- C*-algebra
- multiplier
- inner corona