Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-27T03:35:27.539Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  19 September 2016

Robert J. Archbold
Affiliation:
Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, UK (r.archbold@abdn.ac.uk; somerset@quidinish.fsnet.co.uk)
Douglas W. B. Somerset
Affiliation:
Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, UK (r.archbold@abdn.ac.uk; somerset@quidinish.fsnet.co.uk)

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 = β\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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)