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Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

  • Michael Brannan,
    Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
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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 $N \to \infty$. 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 $N \to \infty$ 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 $L^2$-$L^\infty$ norm equivalence for non-commutative polynomials in free semicircular systems.
Keywords: quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay
MSC Classifications: 46L54, 20G42, 46L65 show english descriptions Free probability and free operator algebras
Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]
Quantizations, deformations
46L54 - Free probability and free operator algebras
20G42 - Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]
46L65 - Quantizations, deformations
 

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