Let $(M,\varphi)=(M_1,\varphi_1)*(M_2,\varphi_2)$ be a free product of non-trivial von Neumann algebras equipped with arbitrary faithful normal states. In 2011, Ueda showed that the diffuse summand of $M$ is a factor if and only if the maximum vector space dimension of $M_1$ and $M_2$ is at least 3, and in this case one further has that the diffuse summand has no asymptotically central sequences. Now consider the von Neumann algebra $N$ generated by non-trivial subalgebras $N_i\leq M_i$ with $\varphi_i$-preserving expectations for $i=1,2$, and let $z \in N$ be the central projection supporting the diffuse summand of $N$. Ueda's result characterizes when $Nz$ is a (full) factor, and, in fact, several cases treated in his proof yield the stronger property that $zMz$ lacks asymptotically $(Nz)$-central sequences. Thus, is natural to wonder: is this stronger property equivalent to the maximum vector space dimension of $N_1$ and $N_2$ being at least 3? In this talk, I will make use of amalgamated free products to answer this question in the affirmative. These same techniques can also be used to analyze the centralizer $M^\varphi = \{ x \in M \colon \varphi(xy) = \varphi(yx)\ \forall y\in M\}$ of the free product state, generalizing a separate 2011 result of Ueda from almost periodic states to arbitrary states. This is based on joint work with Aldo Garcia Guinto, Fehmi Ekin Giritlioglu, Yoonkyeong Lee, and Rahul K. Ramachandran.
