Alphabeta MathModular automorphisms and physical-time KMS conversion
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Analyticity of Holomorphic Functions; Liouville and Morera
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- Banach Alaoglu Goldstine and Krein Milman
- Banach Algebras Spectrum and Holomorphic Functional Calculus
- Banach Valued Integration and the Radon Nikodym Property
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Bounded Variation and the Riemann–Stieltjes Integral
- Compact Operators and Riesz Schauder Theory
- Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Lp Spaces and Test-Function Conventions
- Complex Norms and Realification
- Complex Power Series and Analytic Functions
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Continuous Functional Calculus for Self Adjoint and Normal Operators
- Contour Integration
- Convergence: Nets and Filters
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Cyclic Groups and Direct Products
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Direct Integral Decomposition and Type I Groups
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite Probability and the Probabilistic Method
- Foundations of the Real Numbers for Analysis
- Fourier Transform Convolution and Approximate Identities
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Gelfand Theory and Commutative C Star Algebras
- Geometric Hahn Banach and Convex Separation
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group C Star Algebras and the Fell Unitary Dual
- Group Homomorphisms and the Isomorphism Theorems
- Hausdorff via the Diagonal
- Hilbert Space Geometry and Riesz Representation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Isolated Singularities and Laurent Series
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Local spin matrix algebras and isometric inclusions
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measurable Hilbert Fields and Direct-Integral Operators
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modes of Convergence Egorov and Lusin
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal density states and spectral moments
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Orthonormal Bases, Parseval and Fourier Series
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- Peter Weyl Theory for General Compact Groups
- Physical-time KMS strip equilibrium condition
- pi: the Equivalent Characterizations
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Pure and Mixed States, Ensembles, and Schmidt Decomposition
- Qm Math Composites
- Radon Measures and the Riesz Markov Kakutani Theorem
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Spectral Measures and Borel Functional Calculus
- Splitting Fields
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Suprema and Infima
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- 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 Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorems of Calculus
- The Galois Correspondence
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Residue Theorem and the Evaluation of Real Integrals
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The Winding Number and the Global Cauchy Theorem
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Unbounded Self Adjoint Operators and Stones Theorem
- Uniform Spaces: the Three Definitions
- Uniformly bounded finite-range quantum spin interaction
- Unitary Representations, Positive Type and GNS
- Urysohn's Lemma and the Tietze Extension Theorem
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Von Neumann closure, standard vectors and modular analytic tools
- Weak and Weak Star Topologies
2 · Logical flowchart
3 · Items
Lemma
Spectral analyticity on the modular strip
Lemma
Strong continuity of positive modular unitary powers
Definition
Upper-strip KMS condition for von Neumann algebras
Lemma
Involution and polar identities for the Tomita operator
Definition
Tomita graph and standard modular data
Theorem
Faithful matrix modular flow and Gibbs time
Lemma
Modular resolvent invariance and commutant equality
Theorem