For a fixed C^*-algebra A, we consider all non-commutative dynamical systems that can be generated by A. More precisely, an A-dynamical system is a triple (i,B,\alpha) where \alpha is a *-endomorphism of a C^*-algebra B, and i: A\subseteq B is the inclusion of A as a C^*-subalgebra with the property that B is generated by A\cup\alpha(A)\cup\alpha^2(A)\cup\dotsb. There is a natural hierarchy in the class of A-dynamical systems, and there is a universal one that dominates all others, denoted (i,\mathcal{P}A,\alpha). We establish certain properties of (i,\mathcal{P}A,\alpha) and give applications to some concrete issues of non-commutative dynamics.
For example, we show that every contractive completely positive linear map \varphi:A\to A gives rise to a unique A-dynamical system (i,B,\alpha) that is ‘minimal’ with respect to \varphi, and we show that its C^*-algebra B can be embedded in the multiplier algebra of A\otimes\mathcal{K}.