The Inner Corona Algebra of a C0(X)-Algebra

Robert J. Archbold, Douglas W. B. Somerset

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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 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 languageEnglish
Pages (from-to)299-318
Number of pages20
JournalProceedings of the Edinburgh Mathematical Society
Volume60
Issue number2
Early online date19 Sep 2016
DOIs
Publication statusPublished - May 2017

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Corona
Algebra
Multiplier Algebra
Primitive Ideal
First Countable
Continuum Hypothesis
Closed Ideals
Power set
Compact Hausdorff Space
Lindelöf
Compact Operator
Subalgebra
Injection
Continuous Function
Hilbert space

Keywords

  • C0(X)-algebra
  • C*-algebra
  • multiplier
  • inner corona

Cite this

The Inner Corona Algebra of a C0(X)-Algebra. / Archbold, Robert J.; Somerset, Douglas W. B.

In: Proceedings of the Edinburgh Mathematical Society, Vol. 60, No. 2, 05.2017, p. 299-318.

Research output: Contribution to journalArticle

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