A {\it trace} on a unital C$^*$-algebra $A$ is a state $\varphi:A\rightarrow\mathbb C$ which satisfies the trace property: $\varphi(xy)=\varphi(yx)$, for every $x,y\in A$. The set $\text{T}(A)$ of all traces on $A$ is a convex set which is compact and metrizable, in the weak$^*$-topology. Moreover, it is a metrizable Choquet simplex: every trace can be written uniquely as an ``integral" of extreme traces. The trace simplex is an important, well-studied invariant of C$^*$-algebras. Many C$^*$-algebras of interest, such as reduced C$^*$-algebras of countable groups with trivial amenable radical, have a unique trace and thus a trace space consisting of a single point.
I will present a result showing that for a wide class of C$^*$-algebras $A$ (full free products of unital, separable C$^*$-algebras), the trace space $\text{T}(A)$ is typically a Poulsen simplex, i.e., the extreme traces are dense in $\text{T}(A)$. In particular, the trace space of such C$^*$-algebras $A$ is as large as possible: it contains any metrizable Choquet simplex as a closed face. This result implies that $\text{T}(A_1*A_2)$ is a Poulsen simplex if $A_1$ and $A_2$ are finite dimensional and have no $1$-dimensional direct summands. The proof relies on a new perturbation theorem for pairs of tracial von Neumann algebras $(M_1,M_2)$ which gives necessary conditions ensuring that $M_1$ and a {\it small unitary perturbation} of $M_2$ generate a II$_1$ factor. This is based on joint work with Pieter Spaas and Itamar Vigdorovich.
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