Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

It is known that the normalized standard generators of the free orthogonal quantum group O-N(+) converge in distribution to a free semicircular system as N -> infinity. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator norm of any non-commutative polynomial in the normalized standard generators of O-N(+) converges as N -> infinity 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 L-2-L-infinity norm equivalence for non-commutative polynomials in free semicircular systems.