Representations of C*-dynamical systems implemented by Cuntz families

Given a dynamical system $(A,\text{al})$ where $A$ is a unital $\text{ca}$-algebra and $\text{al}$ 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 $\text{ca}$-algebra they generate, namely the $\text{ca}$-envelope. We then relate each occurrence of the $\text{ca}$-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 $\text{ca}$-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.