### Abstract

Original language | English |
---|---|

Pages (from-to) | 299-318 |

Number of pages | 20 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 60 |

Issue number | 2 |

Early online date | 19 Sep 2016 |

DOIs | |

Publication status | Published - May 2017 |

### Fingerprint

### Keywords

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

### Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*60*(2), 299-318. https://doi.org/10.1017/S0013091516000171

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

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 60, no. 2, pp. 299-318. https://doi.org/10.1017/S0013091516000171

}

TY - JOUR

T1 - The Inner Corona Algebra of a C0(X)-Algebra

AU - Archbold, Robert J.

AU - Somerset, Douglas W. B.

N1 - We are grateful to the referee for a number of helpful comments.

PY - 2017/5

Y1 - 2017/5

N2 - 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.

AB - 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.

KW - C0(X)-algebra

KW - C-algebra

KW - multiplier

KW - inner corona

U2 - 10.1017/S0013091516000171

DO - 10.1017/S0013091516000171

M3 - Article

VL - 60

SP - 299

EP - 318

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 2

ER -