Motivated by Vesterstrom's theorem for the unital case and by recent results for the Dixmier property, we give a number of equivalent conditions (including weak centrality) for a $C^*$-algebra to have the centre-quotient property. We show that every $C^*$-algebra $A$ has a largest weakly central ideal $J_{wc}(A)$. 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 $V_A$ of elements of an arbitrary $C^*$-algebra $A$ which prevent $A$ from having the centre-quotient property. The complement $\mathrm{CQ}(A):= A \setminus V_A$ always contains $Z(A)+J_{wc}(A)$ (where $Z(A)$ is the centre of $A$), with equality if and only if $A/J_{wc}(A)$ is abelian. Otherwise, $\mathrm{CQ}(A)$ fails spectacularly to be a $C^*$-subalgebra of $A$: it is not norm-closed and it is neither closed under addition nor closed under multiplication.

中文翻译：

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C*-代数的中心商性质和弱中心性

受单位情况的 Vesterstrom 定理和 Dixmier 性质的最新结果的启发，我们给出了具有中心商性质的 $C^*$-代数的许多等效条件（包括弱中心性）。我们证明每个 $C^*$-代数 $A$ 都有一个最大的弱中心理想 $J_{wc}(A)$。对于单位 $C^*$-代数 $A$ 的理想 $I$，我们找到了 $A/I$ 的中心元素提升到 $A$ 的中心元素的充分必要条件。这导致了对任意 $C^*$-代数 $A$ 的元素集合 $V_A$ 的表征，这阻止了 $A$ 具有中心商属性。补码 $\mathrm{CQ}(A):= A \setminus V_A$ 总是包含 $Z(A)+J_{wc}(A)$（其中 $Z(A)$ 是 $A$ 的中心） , 相等当且仅当 $A/J_{wc}(A)$ 是阿贝尔。除此以外，