Extensions of $C^*$-Dynamical Systems to Systems with Complete Transfer Operators

Starting from an arbitrary endomorphism $\alpha$ of a unital $C^*$-algebra $\mathcal{A}$ we construct in a canonical way a bigger algebra $\mathcal{B}$ and extend $\alpha$ onto $\mathcal{B}$ in such a way that $\alpha:\mathcal{B} \to \mathcal{B}$ possess a unique non-degenerate transfer operator $\mathcal{L}:\mathcal{B}\to \mathcal{B}$ called complete transfer operator. The pair $(\mathcal{B},\alpha)$ is universal with respect to a suitable notion of a covariant representation and in general depends on a choice of an ideal in $\mathcal{A}$.