Endomorphisms, composition operators and Cuntz families

If b is an inner function and T is the unit circle, then composition with b induces an endomorphism, β, of L¹(T) that leaves H¹(T) invariant. In this document we investigate the structure of the endomorphisms of B(L²(T)) and B(H²(T)) that implement by studying the representations of L¹(T) and H¹(T) in terms of multiplication operators on

B(L²(T)) and B(H²(T)). Our analysis, which was inspired by the work of R. Rochberg and J. McDonald, will range from the theory of composition operators

on spaces of analytic functions to recent work on Cuntz families of isometries and

Hilbert C^*-modules.