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Free random variables whose free distributions are dictated by the semicircular law

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Abstract

In this paper, we study free-probabilistic structures of \( C^{*} \)-algebras generated by mutually free, multi free random variables followed by the semicircular law in a certain sense. Our main results (i) show that a semicircular element in a \( C^{*} \)-probability space \(\left( A,\varphi \right) \), and a certain \( * \)-isomorphism on A generate countable-infinitely many free random variables followed by the semicircular law, (ii) illustrate that, from mutually free, multi free random variables of (i), the corresponding \( C^{*} \)-probability spaces are well-determined, and (iii) characterize not only free-probabilistic structures, but also free-distributional data on the \( C^{*} \)-probability spaces of (ii). As application, we study some types of free random variables followed by the circular law, and followed by free Poisson distributions in certain senses.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, St. Ambrose University, 421 Ambrose Hall, 518 W. Locust St., Davenport, IA, 52803, USA

    Ilwoo Cho & Palle E. T. Jorgensen

  2. Univ. of Iowa, Dept. of Math., 14 McLean Hall, Iowa City, IA, 52242, USA

    Ilwoo Cho & Palle E. T. Jorgensen

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  1. Ilwoo Cho
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  2. Palle E. T. Jorgensen
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Correspondence to Ilwoo Cho.

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Communicated by M. S. Moslehian.

Dedication to the memory of Prof. Derek Robinson.

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Cho, I., Jorgensen, P.E.T. Free random variables whose free distributions are dictated by the semicircular law. Adv. Oper. Theory 9, 34 (2024). https://doi.org/10.1007/s43036-024-00334-9

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