### Abstract

We give a number of equivalent conditions (including weak centrality) for a general C∗-algebra to have the centre-quotient property. We show that every C∗-algebra A has a largest weakly central ideal Jwc(A). For an ideal I of a unital C∗-algebra A, we find a necessary and sufficient condition for a central element of A/I to lift to a central element of A. This leads to a characterisation of the set VA of elements of an arbitrary C∗-algebra A which prevent A from having the centre-quotient property. The complement CQ(A) := A \ VA always contains Z(A) + Jwc(A) (where Z(A) is the centre of A), with equality if and only if A/Jwc(A) is abelian. Otherwise, CQ(A) fails spectacularly to be a C∗-subalgebra of A.

Original language | English |
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Article number | rnaa133 |

Pages (from-to) | 1-44 |

Number of pages | 44 |

Journal | International Mathematics Research Notices |

Early online date | 10 Jun 2020 |

DOIs | |

Publication status | E-pub ahead of print - 10 Jun 2020 |

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## Cite this

Archbold, R. J., & Gogic, I. (2020). The centre-quotient property and weak centrality for C

^{∗}-algebras.*International Mathematics Research Notices*, 1-44. [rnaa133]. https://doi.org/10.1093/imrn/rnaa133