Beschreibung:
<jats:title>Abstract</jats:title><jats:p>It is known that the normalized standard generators of the free orthogonal quantum group<jats:italic>O<jats:sup>+</jats:sup><jats:sub>N</jats:sub></jats:italic>converge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators of<jats:italic>O</jats:italic><jats:sup>+</jats:sup><jats:sub><jats:italic>N</jats:italic></jats:sub>converges as<jats:italic>N</jats:italic>→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-known<jats:italic>L</jats:italic><jats:sup>2</jats:sup>-<jats:italic>L</jats:italic><jats:sup>∞</jats:sup>norm equivalence for noncommutative polynomials in free semicircular systems.</jats:p>