Strong convergence for reduced free products (Remarks on a result by Paul Skoufranis)

Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis's recent result showing that the strong convergence of possibly non-commutative random variables $X^{(k)}\to X$ is stable under reduced free product with a fixed non-commutative random variable $Y$. In fact we obtain a more general fact: assuming that the families $X^{(k)}=\{X_i^{(k)}\}$ and $Y^{(k)}=\{Y_j^{(k)}\}$ are $*$-free as well as their limits (in moments) $X =\{X_i \}$ and $Y =\{Y_j\}$, the strong convergences $X^{(k)}\to X$ and $Y^{(k)}\to Y$ imply that of $\{X^{(k)},Y^{(k)} \}$ to $\{X ,Y\}$. Phrased in more striking language: the reduced free product is "continuous" with respect to strong convergence. The analogue for weak convergence (i.e. convergence of all moments) is obvious. Our approach extends to the amalgamated free product, left open by Skoufranis.