Ja n 20 02 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 Adynamical 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∪α(A)∪α(A)∪ · · · . 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 → A gives rise to to a uniqueA-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⊗K.