Induced Representations of Locally Compact Groups (Cambridge Tracts in Mathematics)

Description

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