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Contemporary Mathematics The Universal A-Dynamical System

Summary: Contemporary Mathematics
The Universal A-Dynamical System
William Arveson
Abstract. For any C-algebra A, an A-dynamical system is a C-dynamical
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 A-dynamical system, emphasizing the role of continuous free prod-
ucts of C-algebras, noncommutative moment polynomials, and conditional
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 A-dynamical 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-


Source: Arveson, William - Department of Mathematics, University of California at Berkeley


Collections: Mathematics