TY - JOUR
ID - 46452
T1 - Non-commutative rational functions in strong convergent random variables
JO - Advances in Operator Theory
JA - AOT
LA - en
SN -
A1 - Yin, Sheng
Y1 - 2018
PY - 2018/01/01
VL - 3
IS - 1
SP - 178
EP - 192
KW - Strong convergence
KW - non-commutative rational functions
KW - random matrices
DO - 10.22034/aot.1702-1126
N2 - 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 $(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)})$ of independent GUE random matrices, $r(X_{1}^{left(nright)},cdots,X_{m}^{left(nright)})$ 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 $r$.
UR - http://www.aot-math.org/article_46452.html
L1 - http://www.aot-math.org/pdf_46452_614d056f5f7799b607d6277111157ff4.html
ER -