Let $A$ be a $C^*$-algebra with centre $Z(A)$. If $I$ is a closed two-sided ideal of $A$, it is immediate that $(Z(A)+I)/I \subseteq Z(A/I)$. A $C^*$-algebra $A$ is said to have the centre-quotient property (shortly, the CQ-property) if for any closed two-sided ideal $I$ of $A$,
$$(Z(A)+I)/I = Z(A/I).$$
By a famous result of Vesterstr\{o}m from 1971, a unital $C^*$-algebra $A$ has the CQ-property if and only if it is weakly central, that is for any pair of maximal ideals $M$ and $N$ of $A$, $M\cap Z(A) =N \cap Z(A)$ implies $M=N$. The most prominent examples of weakly central $C^*$-algebras $A$ are those satisfying the Dixmier property, that is for each $x \in A$ the closure of the convex hull of the unitary orbit of $x$ intersects $Z(A)$. In particular, von Neumann algebras are weakly central.

In this talk we study weak centrality, the CQ-property and several equivalent conditions for general $C^*$-algebras that are not necessarily unital. We then investigate the failure of weak centrality in two different ways. Firstly, we show that every $C^*$-algebra $A$ has a largest weakly central ideal $J_{wc}(A)$, which can be readily determined in several examples. Secondly, we study the set $V_A$ of individual elements of $A$ which prevent the weak centrality (or the CQ-property) of $A$. The set $V_A$ is contained in the complement of $J_{wc}(A)$ and, in certain cases, is somewhat smaller than one might expect. In the course of this, we address a fundamental lifting problem that is closely linked to the CQ-property: for a fixed 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 is a joint work with Robert J. Archbold (University of Aberdeen).