Group Actions and Representations

Abstract

To facilitate the study of abstract objects, a central theme in mathematics is their representations as more concrete ones while preserving their fundamental properties. This procedure is evidently useful if the representing objects have additional special structure, which then allows to carry out concrete calculations in specific situations, or to obtain complete descriptions of the abstract objects under study. One of the greatest achievements of mathematics, the classification of finite simple groups, relies heavily on such representation techniques. But also in this book we have already encountered instances of this phenomenon: In Chapter 4, we studied one-dimensional representations of C ^∗-algebras, i.e., multiplicative linear functionals, and thereby arrived at the Gelfand–Naimark theorem; in Chapter 14 we studied characters, i.e., one-dimensional representations of locally compact Abelian groups (where we left a gap to be filled in this chapter); also the proof of Ellis’ theorem in Appendix G uses representation theory of compact semigroups. We devote this chapter to the fundamentals of representation theory of compact groups. As a by-product we also take a look at actions of compact groups both in the topological and in the measure-preserving settings. All these will be crucial in Chapters 16 and 17 when we return to operators, particularly to Koopman operators of dynamical systems, and apply the developed representation theory to obtain a basic structural description of the dynamical system.