Dynamical Systems and Operator Algebras

We study the algebraic consequences of the traditional descriptions of dynamics based on differential equations. We thus show how C* -dynamical systems consisting of an action of a locally compact group on a C*-algebra provide a common framework for studying the time evolution and symmetry groups of systems in classical and quantum physics. AMS subject classifications. 46L55, 20C35, 81Pl0. One thing which emerged dearly at the workshop in March is that there are many different ideas about what constitutes a "dynamical system." In my talk, therefore, I tried to explain how the dynamical systems operator-algebraists study arise from the more classical notion of dynamical system as a system of ordinary differential equations. In discussions afterwards, I became aware that many workers in my own area are unaware of the circle of ideas that led to the general acceptance of C''dynamical systems as an important field of research. So I have included in this written version a discussion of how these also arise in quantum mechanics. Dynamical systems were originally mathematical formulations of dynamics how physical systems change in time. By exploring the algebraic properties of dynamics, we shall arrive at a notion of dynamical system which provides a common framework for studying time evolutions and symmetry groups in classical mechanics. We then seek a parallel framework for Hilbert-space models of quantum mechanics; carrying out this program raises a variety of interesting mathematical issues. We finish by showing how C*-algebraic dynamical systems bring these ideas together in one neat