Unitary Representations of Groups, Duals, and Characters: (Mathematical Surveys and Monographs)

Unitary representations of groups play an important role in many subjects, including number theory, geometry, probability theory, partial differential equations, and quantum mechanics. This monograph focuses on dual spaces associated to a group, which are spaces of building blocks of general unitary representations. Special attention is paid to discrete groups for which the unitary dual, the most common dual space, has proven to be not useful in general and for which other duals spaces have to be considered, such as the primitive dual, the normal quasi-dual, or spaces of characters. The book offers a detailed exposition of these alternative dual spaces and covers the basic facts about unitary representations and operator algebras needed for their study. Complete and elementary proofs are provided for most of the fundamental results that up to now have been accessible only in original papers and appear here for the first time in textbook form. A special feature of this monograph is that the theory is systematically illustrated by a family of examples of discrete groups for which the various dual spaces are discussed in great detail: infinite dihedral group, Heisenberg groups, affine groups of fields, solvable Baumslag-Solitar group, lamplighter group, and general and special linear groups. The book will appeal to graduate students who wish to learn the basics facts of an important topic and provides a useful resource for researchers from a variety of areas. The only prerequisites are a basic background in group theory, measure theory, and operator algebras.

Table of Contents:
Introduction Unitary dual and primitive dual Representations of locally compact abelian groups Examples of irreducible representations Finite-dimensional irreducible representations Describing all irreducible representations of some semi-direct products Types for representations, quasi-duals, groups of type I Non type I groups Representations of C*-algebras of LC groups, the Glimm Theorem Examples of primitive duals Normal qluasi-dual and characters Finite characters and Thoma's dual Examples of Thoma's duals The group measure space construction Construction of factor representations for some semi-direct products Separating families of finite type representations Appendix Bibliography Notation index Index.

About the Author :
Bachir Bekka, IRMAR, Universite de Rennes 1, France, and Pierre de la Harpe, Universite de Geneve, Switerzerland