Czech Academy of Sciences

The classification by K-theory and traces of the category of simple, separable, nuclear, Z-stable C*-algebras satisfying the UCT is an extraordinary feat of mathematics. What's more, it provides powerful machinery for the analysis of the internal structure of these regular C*-algebras. In this talk, I will explain one such application of classification: In the subclass of classifiable C*-algebras consisting of those for which the simplex of tracial states is nonempty, with extremal boundary that is compact and has the structure of a connected topological manifold, automorphisms can be shown to be generically tracially chaotic. Using similar ideas, I will also show how certain stably projectionless C*-algebras can be described as crossed products.

Let $T_1,\dots,T_n$ be bounded linear operators on a complex Hilbert space

$H$. We study the question whether it is possible to find a unit vector

$x\in H$ such that $|\langle T_jx, x\rangle|$ is large for all $j$. Thus

we are looking for a generalization

of the well-known fact for $n = 1$ that the numerical radius $w(T)$ of a

single operator T satisfies $w(T)\ge \|T\|/2$.