Summary: Given a dynamical system \((A,\alpha)\), where \(A\) is a unital \(C^*\)-algebra and \(\alpha\) is a (possibly nonunital) \(*\)-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 nonselfadjoint operator algebras that depend on the choice of the covariance relation, along with the smallest \(C^*\)-algebra they generate, namely the \(C^*\)-envelope. We then relate each occurrence of the \(C^*\)-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 \(C^*\)-algebras, these results can be interpreted as analogues of Stacey’s famous result, for nonautomorphic 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.