@article {
author = {Yin, Sheng},
title = {Non-commutative rational functions in strong convergent random variables},
journal = {Advances in Operator Theory},
volume = {3},
number = {1},
pages = {178-192},
year = {2018},
publisher = {Tusi Mathematical Research Group (TMRG)},
issn = {2538-225X},
eissn = {2538-225X},
doi = {10.22034/aot.1702-1126},
abstract = {Random matrices like GUE, GOE and GSE have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and ThorbjÃ¸rnsen in their paper in 2005, it is called strong convergence property and then more random matrices with this property are followed. In general, the definition can be stated for a sequence of tuples over some $\text{C}^{\ast}$-algebras. In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. As a direct corollary, we can conclude that for a tuple $(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)})$ of independent GUE random matrices, $r(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)})$ converges in trace and in norm to $r(s_{1},\cdots,s_{m})$ almost surely, where $r$ is a rational function and $(s_{1},\cdots,s_{m})$ is a tuple of freely independent semi-circular elements which lies in the domain of $r$.},
keywords = {Strong convergence,non-commutative rational functions,random matrices},
url = {http://www.aot-math.org/article_46452.html},
eprint = {http://www.aot-math.org/article_46452_614d056f5f7799b607d6277111157ff4.pdf}
}