Advances in Operator Theory

Non-commutative rational functions in strong convergent random variables

Sheng Yin

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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 Thorbørnsen 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 $\mathrm{C}^*$-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 $r$.

Article information

Adv. Oper. Theory Volume 3, Number 1 (2018), 178-192.

Received: 23 February 2017
Accepted: 29 May 2017
First available in Project Euclid: 5 December 2017

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Digital Object Identifier

Primary: 47B06: Riesz operators; eigenvalue distributions; approximation numbers, s- numbers, Kolmogorov numbers, entropy numbers, etc. of operators
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

strong convergence non-commutative rational functions random matrices


Yin, Sheng. Non-commutative rational functions in strong convergent random variables. Adv. Oper. Theory 3 (2018), no. 1, 178--192. doi:10.22034/aot.1702-1126.

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