Strong Asymptotic Freeness for Free Orthogonal Quantum Groups
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© 2014 Canadian Mathematical Society. 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 O→ ∞ 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 O→ ∞ 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 L2-L∞ norm equivalence for noncommutative polynomials in free semicircular systems.
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