Generators of Noncommutative Dynamics

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