ABSTRACT
This chapter aims to construct free (weighted-)Poisson elements of the Banach probability space and to study the homomorphisms acting on LQ induced by certain order-preserving bijective functions h± on the set Z of all integers. It shows that free weighted-Poisson elements are factorized by certain operators and free Poisson elements. More interestingly, distorted free (weighted-)Poisson distributions are studied under the action of 𝔅 on LQ. Free probability is the non-commutative operator-algebraic version of classical measure theory and statistical analysis. The operator-algebraic freeness plays like the classical functional independence by replacing measures on sets to linear functionals on non-commutative algebras. By the central limit theorem, studying semicircular elements whose free distributions are the semicircular law, is one of the main topics in free probability theory.
