Shift and Quasi-Shift Endomorphisms Associated with Representations of Cuntz Algebras

Abstract

We present an overview of the main literature regarding *-endomorphisms of the von Neumann algebra B(H), the bounded linear operators on a separable, in nite- dimensional Hilbert space H. In doing so, we follow the established technique of ex- ploiting the surjective correspondence with non-degenerate representations of Toeplitz- Cuntz algebras. Moreover, we introduce an unstudied class of endomorphism, which we call quasi-shift endomorphisms, and show that they are determined by the well- known class of shift endomorphisms. This is original research adapted from our recent pre-print Quasi-shift Endomorphisms Associated with Representations of Cuntz Alge- bras [13]. i