Alphabeta MathUnitary Representations, Positive Type and GNS
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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
- 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
Complex Analysis
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Complex Norms and Realification
- Complex Subspace Realification
- Continuous Functional Calculus for Self Adjoint and Normal Operators
- Dual Spaces Adjoint Operators and Annihilators
- Gelfand Theory and Commutative C Star Algebras
- Hilbert Space Geometry and Riesz Representation
- Locally Convex Spaces and Continuous Separation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Spectral Measures and Borel Functional Calculus
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
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
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- 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
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- 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
- Sine, Cosine, and the Definition of Pi
- 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 Fundamental Theorems of Calculus
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- 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
Definition
Strongly continuous unitary representations, invariant linear subspaces and intertwiners
Lemma
Continuity criteria for unitary representations
Definition
Cyclic vector and cyclic unitary representation
Theorem
Schur lemma for complex unitary representations
Definition
Matrix coefficient of a unitary representation
Lemma
Bounds and two-sided uniform continuity of unitary coefficients
Definition
Continuous positive-type functions and normalization
Lemma
Diagonal unitary coefficients have positive type
Lemma
Positive-type functions define the GNS pre-Hilbert form
Lemma
The GNS null space is invariant under left translation
Lemma
The GNS translation action is unitary and strongly continuous
Theorem
GNS construction for a continuous positive-type function
Theorem
Uniqueness of the pointed cyclic GNS representation
Corollary
Normalized positive type and pointed cyclic unitary representations
Lemma
Dominated positive type and positive commutant contractions
Lemma
Nonscalar commutant contractions and convex decompositions
Theorem
Extreme normalized positive type is equivalent to irreducible GNS
Sources
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1 and Propositions A.1.2–A.1.4, Appendix A, printed pp. 305–307
- Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, Definition 3.4.1, §3.4, printed p. 106
- Karl-Hermann Neeb, Unitary Representation Theory (2024), §1.2, Lemma 1.2.6, printed p. 15; §1.3, Exercise 1.3.3, printed p. 26
- Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, corrected 2025 notes, §3.4, Proposition 3.4.3, printed pp. 106–107
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition C.4.8, Appendix C, printed p. 375
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Definition 1.B.4, §1.B
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Theorem A.2.2, Appendix A, printed pp. 312–314
- Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, corrected 2025 notes, Proposition 3.4.17 and proof, §3.4, printed pp. 113–115
- Neeb, An Introduction to Unitary Representations of Lie Groups, §§3.4, 4.2 and 5.3
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1, Appendix A, printed p. 305
- Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, §3.4, printed pp. 106–107
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1, Appendix A printed pp. 305–306, and the left/right uniform-continuity convention in §A.3 printed pp. 318–319
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition C.4.1 and Proposition C.4.2, Appendix C, printed pp. 373–374
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Definition 1.B.1, Chapter 1 §1.B, printed p. 26
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Proposition C.4.3 and Example C.1.3, Appendix C, printed pp. 362 and 374–375
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, §1.B
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Construction 1.B.5, Chapter 1 §1.B, printed pp. 27–28
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Appendix C §C.4, proof of Theorem C.4.10, printed pp. 375–376
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Theorem C.4.10, Appendix C §C.4, printed pp. 376–377
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Theorem C.4.10 (GNS Construction)
- Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Theorem C.4.10 and complete proof
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Proposition 1.B.8 and complete proof
- Bekka and de la Harpe, Unitary Representations of Groups, Duals, and Characters, Proposition 1.B.8 and its injectivity proof
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Proposition C.5.1 and complete proof
- Karl-Hermann Neeb, An Introduction to Unitary Representations of Lie Groups, Proposition 5.1.11 and Theorem 5.1.12 with complete proof
- Bekka, de la Harpe and Valette, Kazhdan's Property (T), Theorem C.5.2 and complete proof