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. Cover Title page Foreword Acknowledgments Introduction Representation theory of finite groups Representations of compact groups and abelian groups Primitive dual and locally compact groups of type I The normal quasi-dual of a locally compact group Characters of a locally compact group Thoma’s dual space Historical comments on dual spaces of groups Overview Chapter 1. Unitary dual and primitive dual 1.A. Definition of the unitary dual 1.B. Functions of positive type and GNS construction 1.C. Weak containment and Fell topology for representations of topological groups 1.D. Topological properties of the dual of a group 1.E. Primitive dual of a topological group 1.F. Induced representations, irreducibility, and equivalence 1.G. On decomposing representations into irreducible representations 1.H. On decomposable and diagonalisable operators 1.I. Direct integrals of von Neumann algebras Chapter 2. Representations of locally compact abelian groups 2.A. Canonical representations of abelian groups 2.B. Canonical decomposition of representations of abelian groups 2.C. The SNAG Theorem 2.D. Containment and weak containment in terms of projection-valued measures 2.E. Canonical decomposition of projection-valued measures 2.F. Applying the SNAG Theorem to nonabelian groups Chapter 3. Examples of irreducible representations 3.A. Infinite dihedral group 3.B. Two-step nilpotent groups 3.C. The affine group of a field 3.D. Solvable Baumslag–Solitar groups 3.E. Lamplighter group 3.F. General linear groups 3.G. Some nondiscrete examples Chapter 4. Finite-dimensional irreducible representations 4.A. Finite-dimensional irreducible representations of some semi-direct products 4.B. All finite-dimensional irreducible representations for some groups 4.C. Classes of groups in terms of finite-dimensional representations Chapter 5. Describing all irreducible representations of some semi-direct products 5.A. Constructing some irreducible representations 5.B. Cocycles over measurable group actions 5.C. Cocycles with values in the unitary group of a Hilbert space 5.D. Constructing all irreducible representations 5.E. Identifying induced representations 5.F. On the existence of infinite nonatomic ergodic invariant measures Chapter 6. Types for representations, quasi-duals, groups of type I 6.A. Comparing representations, quasi-equivalence 6.B. Group representations and von Neumann algebras 6.C. The quasi-dual of a topological group 6.D. Groups of type I 6.E. A class of groups of type I Chapter 7. Non type I groups 7.A. A class of non type I groups 7.B. Operator algebras, traces, and types 7.C. Types of group representations 7.D. Types of representations of discrete groups 7.E. On types of representations of locally compact groups 7.F. Non type I factor representations and irreducible representations Chapter 8. Representations of C*-algebras and of LC groups, the Glimm Theorem 8.A. Spectrum and primitive ideal space of a C*-algebra 8.B. C*-algebras and representations of LC groups 8.C. Functorial properties of group C*-algebras 8.D. Second-countable and σ-compact LC groups 8.E. The central character of a representation, of a primitive ideal 8.F. Characterization of type I groups: The Glimm Theorem 8.G. The von Neumann algebra of a group representation 8.H. Variants Chapter 9. Examples of primitive duals 9.A. A weak containment result for induced representations 9.B. Two-step nilpotent groups 9.C. Affine groups of infinite fields 9.D. Solvable Baumslag–Solitar groups 9.E. Lamplighter group 9.F. General and special linear groups 9.G. On the noninjectivity of the map from the dual to the primitive dual 9.H. Borel comparison for duals of groups Chapter 10. Normal quasi-dual and characters 10.A. Traces and Hilbert algebras 10.B. The standard representation of a semi-finite von Neumann algebra 10.C. Group representations associated with traces 10.D. Normal factor representations and characters Chapter 11. Finite characters and Thoma’s dual 11.A. Factor representations of finite type and finite characters 11.B. GNS Construction for traces on groups 11.C. Thoma’s dual 11.D. Characters and primitive duals Chapter 12. Examples of Thoma’s duals 12.A. Two-step nilpotent groups 12.B. Affine groups of infinite fields 12.C. Solvable Baumslag–Solitar groups 12.D. Lamplighter group 12.E. General linear groups Chapter 13. The group measure space construction 13.A. Construction of factors of different types 13.B. Ergodic group actions without invariant measure Chapter 14. Construction of factor representations for some semi-direct products 14.A. Crossed product von Neumann algebras for semi-direct products 14.B. Some factor representations of semi-direct products 14.C. Some normal factor representations 14.D. The von Neumann algebra of the regular representation as a crossed product 14.E. Examples of normal factor representations Chapter 15. Separating families of finite type representations 15.A. Finite type representations and bi-invariant metrics 15.B. Groups for which finite type representations are separating 15.C. Locally compact groups with a finite von Neumann algebra 15.D. Connected and totally disconnected LC groups 15.E. Faithful traces on group C*-algebras 15.F. Traces and Invariant Random Subgroups Appendix A.1. Topology A.2. Borel spaces A.3. Measures on Borel spaces and σ-finiteness A.4. Radon measures on locally compact spaces A.5. Groups and actions A.6. Locally compact groups A.7. Locally compact abelian groups and duality A.8. Hilbert spaces and operators A.9. Projection-valued measures A.10. C*-algebras A.11. Von Neumann algebras Bibliography Notation Index Index Back Cover "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" -- Publisher's description Unitary representations of groups play an important role in many subjects, including number theory, geometry, probability theory, partial differential equations, and quantum mechanics. This book focuses on dual spaces associated to a group, which are spaces of building blocks of general unitary representations.