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

Part of: Selfadjoint operator algebras Peculiar spaces Maps and general types of spaces defined by maps Linear function spaces and their duals Fairly general properties Commutative Banach algebras and commutative topological algebras Generalities

Published online by Cambridge University Press:  19 September 2016

Robert J. Archbold  and
Douglas W. B. Somerset
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)
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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.

Keywords

C 0(X)-algebraC*-algebramultiplierinner corona

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L08: $C^*$-modules 46L45: Decomposition theory for $C^*$-algebras 46E25: Rings and algebras of continuous, differentiable or analytic functions 46J10: Banach algebras of continuous functions, function algebras 54A35: Consistency and independence results 54C35: Function spaces 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.) 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54G10: $P$-spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 2 , May 2017 , pp. 299 - 318
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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