Dynamics and C*-algebras

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Operator algebras were initially studied as models for quantum mechanical systems, because the noncommutativity of multiplication of operators can be used to encode the Heisenberg uncertaintly principle. Since then, this same noncommutative nature of operator algebras has been applied in many other areas of mathematics. This has been particularly successful for dynamical systems, the mathematical models for the spatial and temporal evolution of physical systems. This interaction between operator algebras and dynamical systems is the subject of my research.******There are quite general constructions of operator algebras from dynamical systems. In the view of Alain Connes' program of noncommutative geometry, these act as a replacement for the space of orbits of the system. This first provides a range of tools for understanding the dynamics through their associated operator algebras. This has been particularly effective over the past thirty years, as Connes' program has provided powerful new ideas. As an example, Krieger gave new invariants for symbolic systems through K-theory of the C*-algebras. Results with my co-authors show how analogous results can provide complete invariants for the orbit structure of other systems. ***The construction from dynamical systems also provides a rich source of examples of operator algebras and understanding their structure has been a major goal for the field. George Elliott's classification program for amenable C*-algebras has been one of the largest areas of operator algebras over the past twenty-five years and many of the most impressive results, such as those of Toms and Winter, have been for examples arising from dynamical systems. My own work in this area in the early days produced ideas and technical tools which are still in use today.******My proposal is to continue my investigations into these interactions. There is special emphasis on chaotic systems for which I have recently extended Krieger's invariant for to a much broader class of chaotic dynamical systems. This gives innovative tools for the study of the geometry of fractals. The goal is a better understanding of the invariant but also the development of tools within C*-algebra theory which elucidate the dynamical structure.******In another direction, my goal is to develop tools that provide quantitative measures for a pair of C*-algebras which, although each is constructed from complicated dynamics, are related to each other in a fairly simple fashion.***********