The Structure of Compact Groups (de Gruyter Studies in Mathematics)

Preface to the Fourth Edition
Preface to the Second and Third Editions
Preface to the First Edition
The Logical Dependence of the Contents
Contents
Chapter 1. Basic Topics and Examples
	Definitions and Elementary Examples
	Actions, Subgroups, Quotient Spaces
	Products of Compact Groups
	Applications to Abelian Groups
	Projective Limits
	Totally Disconnected Compact Groups
	Some Duality Theory
	Postscript
	References for this Chapter―Additional Reading
Chapter 2. The Basic Representation Theory of Compact Groups
	Some Basic Representation Theory for Compact Groups
	The Haar Integral
	Consequences of Haar Measure
	The Main Theorem on Hilbert Modules for Compact Groups
	Postscript
	References for this Chapter―Additional Reading
Chapter 3. The Ideas of Peter and Weyl, Tannaka, Hopf, and Hochschild
	Part 1: The Classical Theorem of Peter and Weyl
		An Excursion into Linear Algebra
		The G-Modules E\'⊗E, Hom(E,E) and Hom(E,E)\'
		The Fine Structure of R(G,????)
	Part 2: The General Theory of G-Modules
		Vector Valued Integration
		The First Application: The Averaging Operator
		Compact Groups Acting on Convex Cones
		More Module Actions, Convolutions
		Complexification of Real Representations
	Part 3: The Weakly Complete Group Algebra
		The Hopf Aspect of Weakly Complete Group Algebras
		The Dual of a Weakly Complete Group Hopf Algebra
		A Principal Structure Theorem of ????[G] for Compact G
		The Spectrum of the ????-Algebra R(G,????)
		The Tannaka-Hochschild Duality
		Compact Abelian Groups
		The Probability Semigroup of a Compact G inside ℝ[G]
	Postscript
	References for this Chapter―Additional Reading
Chapter 4. Characters
	Part 1: Characters of Finite Dimensional Representations
	Part 2: The Structure Theorem of E_{fin}
	Cyclic Modules
	Postscript
	References for this Chapter―Additional Reading
Chapter 5. Linear Lie Groups
	Preliminaries
	The Exponential Function and the Logarithm
	Differentiating the Exponential Function in a Banach Algebra
	Local Groups for the Campbell-Hausdorff Multiplication
	Subgroups of A^{-1} and Linear Lie Groups
	Analytic Groups
	The Intrinsic Exponential Function of a Linear Lie Group
	The Adjoint Representation of a Linear Lie Group
	Subalgebras, Ideals, Lie Subgroups, Normal Lie Subgroups
	Normalizers, Centralizers, Centers
	The Commutator Subgroup
	Forced Continuity of Morphisms between Lie Groups
	Quotients of Linear Lie Groups
	The Topological Splitting Theorem for Normal Vector Subgroups
	Postscript
	References for this Chapter―Additional Reading
Chapter 6. Compact Lie Groups
	Compact Lie Algebras
	The Commutator Subgroup of a Compact Lie Group
	The Structure Theorem for Compact Lie Groups
	Maximal Tori
	The Second Structure Theorem for Connected Compact Lie Groups
	Compact Abelian Lie Groups and their Linear Actions
	Action of a Maximal Torus on the Lie Algebra
	The Weyl Group Revisited
	The Commutator Subgroup of Connected Compact Lie Groups
	On the Automorphism Group of a Compact Lie Group
	Covering Groups of Disconnected Compact Lie Groups
	Auerbach\'s Generation Theorem
	The Topology of Connected Compact Lie Groups
	Postscript
	References for this Chapter|Additional Reading
Chapter 7. Duality for Abelian Topological Groups
	The Compact Open Topology and Hom-Groups
	Local Compactness and Duality of Abelian Topological Groups
	Basic Functorial Aspects of Duality
	The Annihilator Mechanism
	Character Groups of Topological Vector Spaces
	The Exponential Function
	Weil\'s Lemma and Compactly Generated Abelian Groups
	Reducing Locally Compact Abelian Groups to Compact Abelian Groups
	A Major Structure Theorem
	The Duality Theorem
	The Identity Component
	The Weight of Locally Compact Abelian Groups
	Postscript
	References for this Chapter―Additional Reading
Chapter 8. Compact Abelian Groups
	Part 1: Aspects of the Algebraic Structure
		Divisibility, Torsion, Connectivity
		Compact Abelian Groups as Factor Groups
	Part 2: Aspects of the Point Set Topological Structure
		Topological Dimension of Compact Abelian Groups
		Arc Connectivity
		Local Connectivity
		Compact Metric Abelian Groups
	Part 3: Aspects of Algebraic Topology―Homotopy
		Free Compact Abelian Groups
		Homotopy of Compact Abelian Groups
		Exponential Function and Homotopy
		The Fine Structure of Free Compact Abelian Groups
	Part 4: Aspects of Homological Algebra
		Injective, Projective, and Free Compact Abelian Groups
	Part 5: Aspects of Algebraic Topology―Cohomology
		Cohomology of Compact Abelian Groups
	Part 6: Aspects of Set Theory
		Arc Components and Borel Sets
	Postscript
	References for this Chapter―Additional Reading
Chapter 9. The Structure of Compact Groups
	Part 1: The Fundamental Structure Theorems of Compact Groups
		Approximating Compact Groups by Compact Lie Groups
		The Closedness of Commutator Subgroups
		Semisimple Compact Connected Groups
		Maximal Connected Abelian Subgroups
		The Splitting Structure Theorem
		Supplementing the Identity Component
	Part 2: The Structure Theorems for the Exponential Function
		The Exponential Function of Compact Groups
		The Dimension of Compact Groups
		Locally Euclidean Compact Groups Are Compact Lie Groups
	Part 3: The Connectivity Structure of Compact Groups
		Arc Connectivity
		Local Connectivity
		Compact Groups and Indecomposable Continua
	Part 4: Some Homological Algebra for Compact Groups
		The Projective Cover of Connected Compact Groups
	Part 5: The Automorphism Group of Compact Groups
		The Iwasawa Theory of Automorphism Groups
		Simple Compact Groups and the Countable Layer Theorem
		The Structure of Compact FC-Groups
		The Commutativity Degree of a Compact Group
	Postscript
	References for this Chapter―Additional Reading
Chapter 10. Compact Group Actions
	A Preparation Involving Compact Semigroups
	Orbits, Orbit Space, and Isotropy
	Equivariance and Cross Sections
	Triviality of an Action
	Quotient Actions, Totally Disconnected G-Spaces
	Compact Lie Groups Acting on Locally Compact Spaces
	Triviality Theorems for Compact Group Actions
	Split Morphisms
	Actions of Compact Groups and Acyclicity
	Fixed Points of Compact Abelian Group Actions
	Transitive Actions of Compact Groups
	Szenthe\'s Theory of Transitive Actions of Compact Groups
	Postscript
	References for this Chapter―Additional Reading
Chapter 11. The Structure of Free Compact Groups
	The Category Theoretical Background
	Splitting the Identity Component
	The Center of a Free Compact Group
	The Commutator Subgroup of a Free Compact Group
	Freeness Versus Projectivity
	Postscript
	References for this Chapter―Additional Reading
Chapter 12. Cardinal Invariants of Compact Groups
	Suitable sets
	Generating Rank and Density
	The Cardinal Invariants of Connected Compact Groups
	Cardinal Invariants in the Absence of Connectivity
	On the Location of Special Generating Sets
	Postscript
	References for this Chapter―Additional Reading
Appendix 1. Abelian Groups
	Examples
	Free Abelian Groups
	Projective Groups
	Torsion Subgroups
	Pure Subgroups
	Free Quotients
	Divisibility
	Some Homological Algebra
	Exact Sequences
	Whitehead\'s Problem
	Postscript
	References for this Appendix―Additional Reading
Appendix 2. Covering Spaces and Groups
	Covering Spaces and Simple Connectivity
	The Group of Covering Transformations
	Universal Covering Groups
	Groups Generated by Local Groups
	Postscript
	References for this Appendix―Additional Reading
Appendix 3. A Primer of Category Theory
	Categories, Morphisms
	Pointed Categories
	Types of Morphisms
	Functors
	Natural Transformations
	Equivalence of Categories
	Limits
	The Continuity of Adjoints
	The Left Adjoint Existence Theorem
	Commutative Monoidal Categories and their Monoids
	Part 1: The Quintessential Diagram Chase
	Part 2: Connected Graded Commutative Hopf Algebras
	Part 3: Duality of Graded Hopf Algebras
	Part 4: An Application to Compact Monoids
	Part 5: Symmetric Hopf Algebras over ℝ and ℂ
	Postscript
	References for this Appendix―Additional Reading
Appendix 4 Selected Results on Topology and Topological Groups
	The Arc Component Topology
	The Weight of a Topological Space
	Metrizability of Topological Groups
	Duality of Vector Spaces
	Subgroups of Topological Groups
	Wallace\'s Lemma
	Cantor Cubes and Dyadic spaces
	Some Basic Facts on Compact Monoids
	Postscript
	References for this Appendix―Additional Reading
Appendix 5. Measures on Compact Groups
	The Definition of Haar Measure
	The Required Background of Radon Measure Theory
	Product Measures
	The Support of a Measure
	Measures on Compact Groups: Convolution
	Semigroup Theoretical Characterization of Haar Measure
	Idempotent Probability Measures on a Compact Group
	Actions and Product Measures
	Nonmeasurable Subgroups of Compact Groups
	Postscript
	References for this Appendix―Additional Reading
Appendix 6 Well-Ordered Projective Limits, Supercompactness, and Compact Homeomorphism Groups
	Well-ordered Lie chains
	Supercompactness
	Compact Homeomorphism Groups
	Postscript
	References for this Appendix―Additional Reading
Appendix 7. Weakly Complete Topological Vector Spaces
	Character Groups of Topological Vector Spaces
	Finite dimensional topological vector spaces
	Duals of vector spaces
	Weakly complete topological vector spaces
	Duality at Work for Weakly Complete Topological Vector Spaces
	Topological Properties of Weakly Complete Topological Vector Spaces
	Tensor Products
	Pro-Lie Groups
	Weakly Complete Unital Algebras
	The Exponential Function
	Postscript
	References for this Appendix―Additional Reading
References
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Index of Symbols
	A-G
	G-S
	S-Z
Index
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