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Winter 2018 Non-commutative rational functions in strong convergent random variables
Sheng Yin
Adv. Oper. Theory 3(1): 178-192 (Winter 2018). 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 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 $r$.

Citation

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Sheng Yin. "Non-commutative rational functions in strong convergent random variables." Adv. Oper. Theory 3 (1) 178 - 192, Winter 2018. https://doi.org/10.22034/aot.1702-1126

Information

Received: 23 February 2017; Accepted: 29 May 2017; Published: Winter 2018
First available in Project Euclid: 5 December 2017

zbMATH: 06804322
MathSciNet: MR3730345
Digital Object Identifier: 10.22034/aot.1702-1126

Subjects:
Primary: 47B06
Secondary: 60B20

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 1 • Winter 2018
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