In this paper, we study deformation in free probability theory. Our setting is systems of specific Banach-space operators acting on a -algebra 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 . 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 . In particular, we outline how properties of those operators affect the semicircular law.
"Certain -homomorphisms of -algebras and sequences of semicircular elements: A Banach space view." Illinois J. Math. 64 (4) 519 - 567, December 2020. https://doi.org/10.1215/00192082-8720474