Ilja Gogić: The Dixmier property and weak centrality for C*-algebras / Mateo Tomašević: Mixed Jordan-power preservers on matrix algebras
Abstract: A unital C-algebra $A$ is said to have the Dixmier property if, for every $x \in A$, the closed convex hull of the unitary orbit of $x$ intersects the centre $Z(A)$. It is well known that all von Neumann algebras satisfy the Dixmier property. Moreover, any unital C-algebra with the Dixmier property is necessarily weakly central, meaning that for any pair of maximal ideals $M, N \subset A$, the equality $M \cap Z(A) = N \cap Z(A)$ implies $M = N$. A famous result of Vesterstrom (1971) characterises weak centrality by the condition that for every closed two-sided ideal $J \subset A$, one has $Z(A/J) = (Z(A) + J)/J$. Later, Magajna (2008) provided an alternative characterisation of weak centrality in terms of averaging by unital completely positive elementary operators. Weak centrality alone is not sufficient to guarantee the Dixmier property, as even simple C-algebras may fail to satisfy it. In fact, a theorem of Haagerup and Zsid\'o (1984) characterises unital simple C-algebras with the Dixmier property as precisely those admitting at most one tracial state. In this talk, we survey the Dixmier property and weak centrality for C*-algebras, with an emphasis on recent developments. The presentation is based on joint work with Robert J.~Archbold and Leonel Robert.
Abstract: In this talk, we build on the problem of automatic additivity for (Jordan-)multiplicative maps between rings by examining maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ (where $n \ge 2$ and $\mathbb{F}$ is an algebraically closed field of characteristic not $2$) that satisfy the mixed Jordan-power identity $$ \phi(A^{k} \circ B) = \phi(A)^{k} \circ \phi(B), \quad \text{for all } A,B \in M_n(\mathbb{F}), $$ where $\circ$ denotes the normalized Jordan product $A \circ B := \frac{1}{2}(AB + BA)$ and $k \in \mathbb{N}$. We show that such maps $\phi$ are necessarily either constant or additive (and in particular are of the form $\varepsilon\psi(\cdot)$, where $\psi$ is a Jordan ring automorphism of $M_n(\mathbb{F})$ and $\varepsilon$ is a $k$-th root of unity). This is joint work with Ilja Gogić.
Both talks will be live and streamed.
Roman Drnovšek and Primož Moravec.
