Non-commutative rational functions in strong convergent random variables

Document Type: Article dedicated to Uffe Haagerup

Author

Faculty of Mathematics, Saarland University

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

Histroty

• Receive Date: 23 February 2017
• Revise Date: 29 May 2017
• Accept Date: 29 May 2017