location:  Publications → journals → CMB
Abstract view

Published:2014-05-07
Printed: Dec 2014
• Michael Brannan,
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
 Format: LaTeX MathJax PDF

## 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
 MSC Classifications: 46L54 - Free probability and free operator algebras 20G42 - Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] 46L65 - Quantizations, deformations

 top of page | contact us | privacy | site map |