Speaker
Description
A unital $C^*$-algebra $A$ is said to satisfy the Dixmier property if for each element $x\in A$ the closed convex hull of the unitary orbit of $x$ intersects the centre $Z(A)$ of $A$. It is well-known that all von Neumann algebras satisfy the Dixmier property and that any unital $C^*$-algebra $A$ that satisfies the Dixmier property is necessarily weakly central, that is, for any pair of maximal ideals $M_1$ and $M_2$ of $A$, $M_1 \cap Z(A) =M_2 \cap Z(A)$ implies $M_1=M_2$. However, weak centrality is not sufficient to guarantee the Dixmier property, as even simple $C^*$-algebras can fail to satisfy it. In fact, a famous result of Haagerup and Zsido from 1984 states that a unital simple $C^*$-algebra satisfies the Dixmier property if and only if it admits at most one tracial state.
In this talk we shall present the overview of the Dixmier property and weak centrality for $C^*$-algebras with an emphasis on more recent results.
