Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

Canadian mathematical bulletin

Michael Brannan¹ 

[1] Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

关键词: quantum groups;    free probability;    asymptotic free independence;    strong convergence;    property of rapid decay;   

DOI  :  10.4153/CMB-2014-004-9

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

It is known that the normalized standard generators of the freeorthogonal quantum group $O_N^+$ converge in distribution to a freesemicircular system as $N o infty$. In this note, wesubstantially improve this convergence result by proving that, inaddition to distributional convergence, the operator norm of anynon-commutative polynomial in the normalized standard generators of$O_N^+$ converges as $N o infty$ to the operator norm of thecorresponding non-commutative polynomial in a standard freesemicircular system. Analogous strong convergence results are obtainedfor the generators of free unitary quantum groups. As applications ofthese results, we obtain a matrix-coefficient version of our strongconvergence theorem, and we recover a well known $L^2$-$L^infty$ normequivalence for non-commutative polynomials in free semicircularsystems.

【 授权许可】

Unknown