### Abstract

Original language | English |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Quarterly Journal of Mathematics |

Volume | 65 |

Issue number | 1 |

Early online date | 22 Jan 2013 |

DOIs | |

Publication status | Published - 1 Mar 2014 |

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### Cite this

*Quarterly Journal of Mathematics*,

*65*(1), 1-24. https://doi.org/10.1093/qmath/has048

**Spectral synthesis in the multiplier algebra of a C_0(X)-algebra.** / Archbold, Robert J; Somerset, Douglas W. B.

Research output: Contribution to journal › Article

*Quarterly Journal of Mathematics*, vol. 65, no. 1, pp. 1-24. https://doi.org/10.1093/qmath/has048

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TY - JOUR

T1 - Spectral synthesis in the multiplier algebra of a C_0(X)-algebra

AU - Archbold, Robert J

AU - Somerset, Douglas W. B.

N1 - We are grateful to the referee for a number of helpful comments and for pointing out an error in the original proof of Theorem 3.6.

PY - 2014/3/1

Y1 - 2014/3/1

N2 - Let A be a C0(X)-algebra with continuous map φ from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M(A) is a C(β X)-algebra with continuous map Graphic: Prim(M(A)) → β X extending φ. For x ∈ Im(φ), let Jx = ⋂{P ∈ Prim(A): φ(P) = x} and Graphic. Then Graphic, the strict closure of Jx in M(A). Thus, Hx is strictly closed if and only if Graphic, and the ‘spectral synthesis’ question asks when this happens. In this paper, it is shown that, for σ-unital A, Hx is strictly closed for all x ∈ Im(φ) if and only if Jx is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.

AB - Let A be a C0(X)-algebra with continuous map φ from Prim(A), the primitive ideal space of A, to a locally compact Hausdorff space X. Then the multiplier algebra M(A) is a C(β X)-algebra with continuous map Graphic: Prim(M(A)) → β X extending φ. For x ∈ Im(φ), let Jx = ⋂{P ∈ Prim(A): φ(P) = x} and Graphic. Then Graphic, the strict closure of Jx in M(A). Thus, Hx is strictly closed if and only if Graphic, and the ‘spectral synthesis’ question asks when this happens. In this paper, it is shown that, for σ-unital A, Hx is strictly closed for all x ∈ Im(φ) if and only if Jx is locally modular for all x ∈ Im(φ) and φ is a closed map relative to its image. Various related results are obtained.

U2 - 10.1093/qmath/has048

DO - 10.1093/qmath/has048

M3 - Article

VL - 65

SP - 1

EP - 24

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

IS - 1

ER -