 
Summary: Contemporary Mathematics
The Universal ADynamical System
William Arveson
Abstract. For any Calgebra A, an Adynamical system is a Cdynamical
system that contains A and can be generated by the images of A under
the semigroup of nonnegative time endomorphisms. There is a universal A
dynamical system that occupies a position in noncommutative dynamics that
resembles the position of the tangent bundle in commutative dynamics.
We describe an approach to noncommutative dilation theory based on the
universal Adynamical system, emphasizing the role of continuous free prod
ucts of Calgebras, noncommutative moment polynomials, and conditional
expectations.
1. Introduction
This paper gives an exposition of a new approach to the dilation theory of
semigroups of completely positive maps on von Neumann algebras. This approach
is based on the notion of an Adynamical system. These objects provide the C

algebraic structure that underlies much of noncommutative dynamics, whether it
takes place in C
algebra or a von Neumann algebra, independently of issues relat
