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

Keywords

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

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