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

TL;DR

This paper proves that strong convergence of non-commutative random variables is preserved under reduced free products, extending previous results and providing a new proof using Ricard and Xu's inequality, with implications for amalgamated free products.

Contribution

The paper offers a new proof of stability of strong convergence under reduced free products and extends the result to amalgamated free products, enhancing understanding of free probability.

Findings

Strong convergence is stable under reduced free products.

The approach extends to amalgamated free products.

Reduced free product is continuous with respect to strong convergence.

Abstract

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.