Professor Boyer will investigate the representation theory of classical groups associated to operator algebras. The emphasis will be on inductive limits of compact groups, which are the next order of complexity after finite dimensional groups. The analysis to be undertaken will use the framework of approximately finite- dimensional C*-algebras (especially the K-functor and its order structure), dynamical systems and their ergodic measures, and the combinatorics of the approximating groups. One specific objective is to classify the representations of the inductive limit unitary group which generalize the well-known quasi-free representations of the canonical (anti-) commutation relations. This project is part of an effort now underway in many quarters to extend concepts and results long familiar in the study of Lie groups (named after the eminent Norwegian mathematician Sophus Lie) to objects of a much more exotic sort. An example of a Lie group in the usual sense is the group of all rotations of a sphere, where the group operation consists of following one motion by another. Detailed knowledge of the rotation group is extremely helpful in the analysis of any mathematical or physical situation where spherical symmetry is present. It is an important feature of this group that we need only a finite number of parameters (in this case three) to specify any of its members. Likewise, we can consider the rotation group in higher dimensions. No truly new phenomena occur as long as the dimension remains finite, but if we let it increase without bound and so to speak take the limit, we obtain a kind of infinite-dimensional Lie group. Many of the same questions make sense for this inductive limit rotation group as for the ordinary three-dimensional one. Professor Boyer will work on the new methods that are needed to answer them in this more difficult setting.
