Representations of C*-dynamical systems implemented by Cuntz families

TL;DR

This paper investigates operator algebra representations of C*-dynamical systems using Cuntz families, analyzing their associated non-selfadjoint algebras, their C*-envelopes, and connections to crossed products, extending Stacey and Exel's frameworks.

Contribution

It introduces a detailed analysis of representations involving Cuntz families for non-unital endomorphisms, computing their C*-envelopes and relating them to twisted crossed products, extending existing theories.

Findings

Computed non-selfadjoint operator algebras for various covariance relations

Identified the C*-envelope as a full corner of a twisted crossed product

Provided a counterexample illustrating the diversity of these algebras

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.