Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

Source abstract

Abstract 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 → ∞. 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 O + N converges as N → ∞ 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 ∞ norm equivalence for noncommutative polynomials in free semicircular systems.