Alphabeta MathThe Modular Function and L1 Group Algebras
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Differential Geometry
Functional Analysis
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Lemma
Right translation scales left Haar measure
Definition
Modular function of a locally compact group
Theorem
The modular function is a continuous homomorphism
Lemma
Haar change of variables under inversion
Definition
Unimodular locally compact group
Lemma
Counting measure on a discrete group is Haar, Haar measures there are its multiples, and integrals against them are sums
Proposition
Compact, discrete and abelian groups are unimodular
Definition
Complex Haar Lᵖ spaces and compactly supported functions
Definition
Compactly supported convolution on a group
Lemma
Convolution preserves compact support and is associative
Lemma
Submultiplicativity of convolution in the L1 norm
Lemma
Completeness of the complex Haar L1 and L2 spaces and density of Cc
Definition
Convolution on L1 of a locally compact group
Definition
Involution on L1 of a locally compact group
Lemma
The L1 involution is isometric, involutive and reverses convolution
Definition
Banach star-algebra without a required unit
Theorem
L1 of a locally compact group is a Banach star-algebra
Lemma
Strong continuity of left and modular right translations on L1 and L2
Theorem
L1 group algebras have a contractively bounded approximate identity
Proposition
The L1 group algebra has a unit exactly when the group is discrete
Definition
Left and right regular unitary representations of an LCH group
Theorem
The regular representations are unitary, strongly continuous, and the left one is faithful
Sources
- Bekka, de la Harpe and Valette, Kazhdan’s Property (T), Appendix A §§A.3–A.4
- Emmanuel Kowalski, Representation Theory of Groups, §§5.2–5.3
- Lynn Loomis, An Introduction to Abstract Harmonic Analysis, §§30–31
- Anthony W. Knapp, Advanced Real Analysis, VI §2
- Lynn Loomis, An Introduction to Abstract Harmonic Analysis, §§30A–30B