### Abstract

Let A = C(X) ⊗ K(H), where X is an infinite compact Hausdorff space and

K(H) is the algebra of compact operators on a separable, infinite-dimensional Hilbert space. Let As be the norm-closed ideal of the multiplier algebra M(A) consisting of all the strong∗ -continuous functions from X to K(H). Then As/A is the inner corona algebra of A. We identify the space MinPrimal (As) of minimal closed primal ideals in As. If A is separable then MinPrimal (As) is compact and extremally disconnected. Using ultrapowers, we exhibit a faithful family of irreducible representations of As/A and hence show that if every point of X lies in the boundary of a zero set (i.e. if X has no P-points) then the minimal closed primal ideals of As/A are precisely the images under the quotient map of the minimal closed primal ideals of As. The map between MinPrimal (As) and MinPrimal (As/A) need not be continuous, however, and MinPrimal (As/A) is not weakly Lindelof. As an application, it is shown that if X = βN\N then the relation of inseparability on Prim(As/A) is an equivalence relation but not an open equivalence relation.

K(H) is the algebra of compact operators on a separable, infinite-dimensional Hilbert space. Let As be the norm-closed ideal of the multiplier algebra M(A) consisting of all the strong∗ -continuous functions from X to K(H). Then As/A is the inner corona algebra of A. We identify the space MinPrimal (As) of minimal closed primal ideals in As. If A is separable then MinPrimal (As) is compact and extremally disconnected. Using ultrapowers, we exhibit a faithful family of irreducible representations of As/A and hence show that if every point of X lies in the boundary of a zero set (i.e. if X has no P-points) then the minimal closed primal ideals of As/A are precisely the images under the quotient map of the minimal closed primal ideals of As. The map between MinPrimal (As) and MinPrimal (As/A) need not be continuous, however, and MinPrimal (As/A) is not weakly Lindelof. As an application, it is shown that if X = βN\N then the relation of inseparability on Prim(As/A) is an equivalence relation but not an open equivalence relation.

Original language | English |
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Pages (from-to) | 117-150 |

Number of pages | 24 |

Journal | Houston Journal of Mathematics |

Volume | 46 |

Issue number | 1 |

Publication status | Published - 2020 |

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## Cite this

Archbold, R. J., & Somerset, D. W. B. (2020). Minimal primal ideals in the inner corona algebra of a C_{0}(X)-algebra.

*Houston Journal of Mathematics*,*46*(1), 117-150.