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∗-algebra Xφ 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 ∗-homomorphisms on Xφ. 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 Xφ. In particular, we outline how properties of those operators affect the semicircular law.