Dynamics and C*-algebras

点击翻译按钮获取中文摘要

My research concerns the fields of operator algebras and dynamical systems and their interplay. Mathematicians model classical mechanical systems with two objects: the collection of possible states of the system (or phase space) and their physically observable quantities. The former is a geometric object and the latter, an algebraic one. The link is that the latter are seen as functions defined on the former and as such their multiplication is a commutative algebraic operation. In passing to quantum mechanical systems, the functions are replaced by operators, which are more general and allow for noncommutative multiplication. A consequence is that the geometric notion of phase space is lost. However, the viewpoint of Alain Connes' program of noncommutative geometry is that many natural geometric concepts may be effectively carried over to the setting of the algebra of operators and give new, geometric insights. Beyond their original use as models in physics, there are now many mathematical objects where relevant operator algebras may be constructed and the tools of noncommutative geometry used to give new understanding to the original object. My main interest is in the example of dynamical systems, which is the study of models of the evolution of physical systems and has a wide variety of applications, including in information technology and coding theory. Since the early days of operator algebras, there has been a lively and deep exchange between these two subjects. On the one hand, the operator algebras constructed from dynamical systems have provided fundamental examples that shaped the subject. In the other direction, the tools of investigation of operator algebras have revealed surprising new insights and invariants for the study of the dynamics.