Banach-Space Operators Acting on Semicircular Elements Induced by Orthogonal Projections

Abstract

The main purposes of this paper are (i) to construct-and-study weighted-semicircular elements from mutually orthogonal $\left| \mathbb {Z} \right|$-many projections, and the Banach $*$-probability space $\mathbb {L}_{Q}$ generated by these operators, (ii) to establish $*$-isomorphisms on $\mathbb {L}_{Q}$ induced by shifting processes on the set $\mathbb {Z}$ of integers, (iii) to consider how the $*$-isomorphisms of (ii) generates Banach-space adjointable operators acting on the Banach $*$-algebra $\mathbb {L}_{Q}$, (iv) to investigate operator-theoretic properties of the operators of (iii), and (v) to study how the Banach-space operators of (iii) distorts the original free-distributional data on $\mathbb {L}_{Q}$. As application, one can check how the semicircular law is distorted by our Banach-space operators on $\mathbb {L}_{Q}$.