Comparing Banach Spaces for Systems of Free Random Variables Followed by the Semicircular Law

Abstract

AbstractWe study certain Banach-space operators from noncommutative free probability, acting on systems of free random variables whose free distributions are followed by the semicircular law. In particular, we consider (i) non-self-adjoint free random variables T of a C ∗-probability space followed by the semicircular law in the sense that: all n-th joint free moments of \(\left \{ T,T^{*}\right \} \) are identical to the \(\frac {n}{2}\)-th Catalan numbers \({c_{\frac {n}{2}}}\), for all \({n\in \mathbb {N}}\), with axiomatization: \(c_{\frac {n}{2}}=0\), whenever \(\frac {n}{2}\notin \mathbb {N}\), (ii) some structure theorems of C ∗-probability spaces generated by countable-infinitely many free random variables followed by the semicircular law, and (iii) certain Banach-space operators acting on free random variables of (i) and (ii).KeywordsFree probabilitySemicircular elementsFree random variables followed by the semicircular lawBanach-space operators