Generators of Noncommutative Dynamics

Abstract

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,α) where $α$ 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 $A\cup α(A)\cup α^2(A)\cup...$. 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,α). We establish certain properties of $(i,PA,α)$ and give applications to some concrete issues of noncommutative dynamics. For example, we show that every contractive completely positive linear map $ϕ: A\to A$ gives rise to to a unique A-dynamical system (i,B,α) that is "minimal" with respect to $ϕ$, and we show that its C*-algebra B can be embedded in the multiplier algebra of $A\otimes {\mathcal K}$.