Strong convergence for reduced free products

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 [Formula: see text] is stable under reduced free product with a fixed non-commutative random variable [Formula: see text]. In fact we obtain a more general fact: assuming that the families [Formula: see text] and [Formula: see text] are ∗-free as well as their limits (in moments) [Formula: see text] and [Formula: see text], the strong convergences [Formula: see text] and [Formula: see text] imply that of [Formula: see text] to [Formula: see text]. 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.