Representations of C*-dynamical systems implemented by Cuntz families

Abstract

Given a dynamical system (A,\al) where $A$ is a unital \ca-algebra and \al is a (possibly non-unital) *-endomorphism of $A$, we examine families $(\pi,\{T_i\})$ such that $\pi$ is a representation of $A$, $\{T_i\}$ is a Toeplitz-Cuntz family and a covariance relation holds. We compute a variety of non-selfadjoint operator algebras that depend on the choice of the covariance relation, along with the smallest \ca-algebra they generate, namely the \ca-envelope. We then relate each occurrence of the \ca-envelope to (a full corner of) an appropriate twisted crossed product. We provide a counterexample to show the extent of this variety. In the context of \ca-algebras, these results can be interpreted as analogues of Stacey's famous result, for non-automorphic systems and $n>1$. Our study involves also the one variable generalized crossed products of Stacey and Exel. In particular, we refine a result that appears in the pioneering paper of Exel on (what is now known as) Exel systems.Comment: 29 pages; changes in subsection 1.2; close to publicatio