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

Abstract

The main purposes of this paper are (i) to construct-and-study (weighted-)semicircular elements induced by 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 $\mathbb {L}_{Q}$ from shifting processes on the set $\mathbb {Z}$ of integers, (iii) to consider the $*$-isomorphisms of (ii) as Banach-space operators acting on $\mathbb {L}_{Q}$ (by regarding the Banach $*$-algebra $\mathbb {L}_{Q}$ as a Banach space), and (iv) to compare the free-distributional data affected by the operators of (iii) from the original data.