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Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

Published online by Cambridge University Press:  20 November 2018

Michael Brannan
Michael Brannan*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA e-mail: [email protected]
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Abstract

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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.

Keywords

46L5420G4246L65quantum groupsfree probabilityasymptotic free independencestrong convergenceproperty of rapid decay
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 4 , 01 December 2014 , pp. 708 - 720
Copyright
Copyright © Canadian Mathematical Society 2014

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