Certain ∗-homomorphisms of C∗-algebras and sequences of semicircular elements: A Banach space view

Abstract

In this paper, we study deformation in free probability theory. Our setting is systems of specific Banach-space operators acting on a $C^{\ast }$-algebra ${\mathcal{X}}_{\phi }$ generated by mutually free, infinitely-many semicircular elements. In particular, we give a constructive classification for the cases where those operators are generated by certain $\ast$-homomorphisms on ${\mathcal{X}}_{\phi }$. Our main results not only classify and characterize specific types of the operators, but they also show how such operators deform the free probability on ${\mathcal{X}}_{\phi }$. In particular, we outline how properties of those operators affect the semicircular law.