Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

Brannan, Michael

Canadian mathematical bulletin, Tome 57 (2014) no. 4, pp. 708-720

Résumé

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 $N\,\to \,\infty$ . 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 $N\,\to \,\infty$ 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 ${{\mathcal{L}}^{2}}\,-\,{{\mathcal{L}}^{\infty }}$ norm equivalence for noncommutative polynomials in free semicircular systems.

Détail

DOI : 10.4153/CMB-2014-004-9

Mots-clés : 46L54, 20G42, 46L65, quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay