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 $$\mathbb {L}_{Q}$$ .