Induced Representations of Locally Compact Groups Cambridge Tracts in Mathematics Hardcover

Locally compact groups and their representation theory underlie areas such as relativity, quantum field theory, signal processing and medical imaging. All who are interested in the foundations of such subjects, as well as pure mathematicians working in harmonic analysis, will appreciate this comprehensive monograph. Contents Preface 1 Basics 1.1 Locally compact groups 1.2 Examples 1.3 Coset spaces and quasi-invariant measures 1.4 Representations 1.5 Representations of L1(G) and functions of positive type 1.6 C*-algebras and weak containment of representations 1.7 Abelian locally compact groups 1.8 Notes and references 2 Induced representations 2.1 Inducing from an open subgroup 2.2 Conditions for irreducibility 2.3 The induced representation in general 2.4 Other realizations Summary Realization I Realization II Realization III Realization III for Semidirect Products 2.5 The affine group and SL(2,R) 2.6 Some basic properties of induced representations 2.7 Induction in stages 2.8 Tensor products of induced representations 2.9 Frobenius reciprocity 2.10 Notes and references 3 The imprimitivity theorem 3.1 Systems of imprimitivity 3.2 Induced systems of imprimitivity 3.3 The imprimitivity theorem 3.4 Proof of the imprimitivity theorem: the general case 3.5 Notes and references 4 Mackey analysis 4.1 Mackey analysis for almost abelian groups 4.2 Orbits in the dual of an abelian normal subgroup 4.3 Mackey analysis for abelian normal subgroups 4.4 Examples: some solvable groups 4.5 Examples: action by compact groups 4.6 Limitations on Mackey's theory 4.7 Cocycles and cocycle representations 4.8 Mackey's theory for a nonabelian normal subgroup 4.9 Notes and references 5 Topologies on dual spaces 5.1 The inner hull-kernel topology 5.2 The subgroup C*-algebra 5.3 The subgroup representation topology and functions of positive type 5.4 Continuity of inducing and restricting representations 5.5 Examples: nilpotent and solvable groups 5.6 The topology on the dual of a motion group 5.7 Examples: motion groups 5.8 The primitive ideal space of a two-step nilpotent group 5.9 Notes and references 6 Topological Frobenius properties 6.1 Amenability and induced representations 6.2 Basic definitions and inheritance properties 6.3 Motion groups 6.4 Property (FP) for discrete groups 6.5 Nilpotent groups 6.6 Notes and references 7 Further applications 7.1 Asymptotic properties of irreducible representations of motion groups 7.2 Projections in L1(G) 7.3 Generalizations of the wavelet transform 7.4 Notes and references Bibliography Index "Locally compact groups arise in many diverse areas of mathematics, the physical sciences, and engineering and the presence of the group is usually felt through unitary representations of the group. This observation underlies the importance of understanding such representations and how they may be constructed, combined, or decomposed. Of particular importance are the irreducible unitary representations. In the middle of the last century, G.W. Mackey initiated a program to develop a systematic method for identifying all the irreducible unitary representations of a given locally compact group G. We denote the set of all unitary equivalence classes of irreducible unitary representations of G by G. Mackey's methods are only effective when G has certain restrictive structural characteristics; nevertheless, time has shown that many of the groups that arise in important problems are appropriate for Mackey's approach. The program Mackey initiated received contributions from many researchers with some of the most substantial advances made by R.J. Blattner and J.M.G. Fell. Fell'swork is particularly important in studying Gas a topological space. At the core of this program is the inducing construction, which is a method of building a unitary representation of a group from a representation of a subgroup"-- Provided by publisher The dual space of a locally compact group G consists of the equivalence classes of irreducible unitary representations of G. This book provides a comprehensive guide to the theory of induced representations and explains its use in describing the dual spaces for important classes of groups. It introduces various induction constructions and proves the core theorems on induced representations, including the fundamental imprimitivity theorem of Mackey and Blattner. An extensive introduction to Mackey analysis is applied to compute dual spaces for a wide variety of examples. Fell's contributions to understanding the natural topology on the dual are also presented. In the final two chapters, the theory is applied in a variety of settings including topological Frobenius properties and continuous wavelet transforms. This book will be useful to graduate students seeking to enter the area as well as experts who need the theory of unitary group representations in their research.