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

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.