Structure of crossed products by amenable groups and classification of group actions

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The main focus of this project is on dynamical systems. Probably the most important example is the time evolution of a physical system. Given complete information about its state at a particular time, one should be able to predict the state at any later time. If the dynamics is "reversible" (such as in Newtonian mechanics), one can also determine the state at any previous time. If there are no outside influences on the system, the new state depends only on the beginning state and how long the system was allowed to run, not on the particular time at which it started. (If you plant a seed in the fall, the result six months later will be different from the result if you plant it in the spring. However, if the climate were the same year round, the result six months later would be the same regardless of when you plant.) This setup is an example of an action of the group of real numbers: the set of possible lengths of time the system is allowed to run. Actions of other groups are also important. The group of integers corresponds to taking time in discrete steps, which is sometimes appropriate and is mathematically simpler. Other groups arise from symmetry in a system or from entirely different considerations. One main goal of this project is to better understand certain kinds of dynamical systems, at this stage primarily discrete ones but for which the groups can be otherwise complicated. One direction is to understand the relation between the system and a particular algebraic object, the so-called crossed product, constructed from the system. Another direction is the complete description in a special case (in which the collection of objects of the system is very complicated but the group is actually finite) of all possible systems.The largest component of this project involves relating the properties of a minimal and essentially free action of a countable amenable group on a compact metric space to the structure of its crossed product. The plan is to push forward the boundaries of our knowledge in two two particular directions: (1) relating the mean dimension of a minimal homeomorphism to the radius of comparison of the crossed product (simplest group, namely, the integers, and most complicated space) and (2) proving regularity properties when the group is arbitrary but the space is the Cantor set (simplest space, most complicated group). It is also hoped that related methods will help with actions on simple C*-algebras when the action has the tracial Rokhlin property. The next component is the classification of pointwise outer actions of finite groups on purely infinite simple separable nuclear C*-algebras, a much larger class of actions than the ones already classified (those with the Rokhlin property). The third component is the study of operator algebras on Lebesgue spaces with exponent different from two. This is a new field which turns out to be unexpectedly rich.