Prerequisite proofs

The selected lessons state which results they use. The links below include programme proofs and external research materials. An external reference does not replace a programme proof. The vector-valued integration and tensor identification needed here are proved in the first lesson.

Measure and integration

• Haar measure, modular function, inversion and compact support estimates
• Measure and Hilbert space tools for Haar integration

C*-Fourier and Plancherel proofs

• Theorem 2.1: C*(G)=C*_r(G)=C0(dual G), both norms sup|Fourier f|
• Theorem 2.1: complete continuous character spectrum of L1(G), compact-open topology
• Theorem 3.2: uniform Fourier-image density in C0(dual G)
• Lemma 1.1: individual L1 vector carrier and L1-valued convolution integral
• Proposition 1.1: finite complex Radon measures determined by positive-sign character transforms
• Theorem 2.1: continuous positive-type function has unique finite positive representing measure
• Theorem 2.1: dual Haar construction and B1 inversion
• Proposition 3.1: primal Haar c dx gives reciprocal dual c^-1 dxi
• Theorem 1.1: onto L2 Fourier unitary agreeing with the integral on L1 intersect L2
• Proposition 2.1(1): L1 convolution on L2 conjugates to Fourier multiplication
• Lemma 2.2: inverse integral on L1 intersect L2 of the dual
• Theorem 3.1: Fourier algebra equals L2*L2 as actual continuous functions
• Corollary 3.3: compact continuous convolution is transform of B1 vector
• Theorem 2.1: evaluation alpha G is a topological group isomorphism onto the bidual
• Proposition 3.1: bidual Haar equals pushforward dx and Fdual F = reflection
• Corollary 4.2: Fourier injectivity on finite Radon measures and L1
• Lemma 5.1: each L2 vector has a sigma-compact open subgroup carrier
• Vector integration lemma VI1-VI3: Bochner norm/Fubini and onto L2(G) tensor K identification for arbitrary K
• Theorem 5.1(4)-(7): Radon-product Tonelli/Fubini at support qualifications
• Theorem 6.2: onto Radon-product L2 Hilbert tensor unitary
• Proposition 3.1(4): Cc density in finite-p Haar spaces
• Proposition 7.2: open sigma-compact subgroup from a compact identity neighborhood
• Proposition 13.7: canonical finite-p comparison, with differing global measures
• Theorems 2.1-2.2: sequence convergence without countability hypotheses
• Theorems 3.1-3.2: Holder, Minkowski, completeness and simple approximation

Spectral and von Neumann foundations for logarithmic products

• Emmanuel Kowalski: Spectral theory in Hilbert spaces. Theorem 3.1, Proposition 4.38, Theorem 4.42 and Corollary 4.43; the lesson supplies the arbitrary-Hilbert-space extension.
• Orr Shalit: Introduction to von Neumann algebras, Lecture 7. Definitions 8–10 and Theorem 11, for the concrete predual and vector-series topology.
• Vaughan F. R. Jones: Von Neumann algebras. Theorem 3.2.2 and Corollary 3.2.3, for the general bicommutant theorem.

Spectral and complex-analysis foundations for power strips

• Emmanuel Kowalski: Spectral theory in Hilbert spaces. Theorems 3.1 and 4.42: the separable spectral theorem, with the arbitrary-Hilbert reduction proved in the lesson.
• Alexandre Vizeff: Complex analysis, Lecture 6. Section 3: bounded-domain harmonic maximum principle.
• Alexandre Vizeff: Complex analysis, Lecture 9. Cauchy integral theorem and formula, with positively oriented contours.
• Alexandre Vizeff: Complex analysis, Lecture 10. Cauchy estimates for Taylor coefficients.
• Alexandre Vizeff: Complex analysis, Lecture 11. Rectangular Morera theorem and proof.

Derivation and innerness foundations

• Section 3, adjoining an identity and its complete C*-norm
• Section 2, Neumann series and inverse estimates; preceding absolute-product lemma
• CF Section 1: explicit choice convention, maximal principle and norm Hahn–Banach proofs
• Theorems 2.1–2.2 and Corollary 2.3; real and complex Hahn–Banach
• Theorem 3.1; complete-metric Baire
• Lemma 4.1 and Theorem 4.2; Banach operator space and arbitrary-family uniform boundedness
• Theorem 5.1, Corollary 5.2 and Theorem 5.3; open mapping, inverse and closed graph
• Theorem 3.1, using Tychonoff Theorem 2.3 in preceding section
• Theorems 5.1 and 5.3; continuous calculus and nonunital form
• Propositions 7.2–7.3; Jordan decomposition and unitary spans; positive cone is in §8
• Theorem 11.4 and Corollary 11.5; arbitrary approximate identity nets
• Theorem 3.1, including (5) commutation with the commutant
• Theorem 4.4; bounded selfadjoint spectral measure and uniqueness
• Theorem 6.1; arbitrary bounded monotone nets
• Theorem 8.1; normal/unitary bounded spectral calculus for L34 small corners
• Theorem 4.4; bicommutant characterization
• Proposition 7.2; polar decomposition inside a von Neumann algebra
• Theorem 9.4; predual of an arbitrary sigma-weakly closed subspace
• Corollaries 11.4–11.5 and following positive-map criterion
• Proposition 3.1, Proposition 3.5, Lemma 4.2; arbitrary joins, central supports, left supports
• Corollary 2.8; Hermitian normal-functional decomposition

Faithful-state core foundations

• OA-FLOW course project: Local Fourier and closed-form foundations for the state core Local proof with its stated mathematical prerequisites.
• OA-FLOW course project: Faithful normal GNS representations and their topologies Local proof with its stated mathematical prerequisites.
• OA-FLOW course project: Full left-Hilbert-algebra domains and the unchanged closed involution Local proof with its stated mathematical prerequisites.
• OA-FLOW course project: MF-06: narrow scalar and vector integration bridge Local proof with its stated mathematical prerequisites.
• OA-FLOW course project: Modular covariance through positive multipliers and operator averages Local proof with its stated mathematical prerequisites.

Operator and Banach foundations

• 1. Positive semidefinite forms
• 2. The projection theorem and the Riesz–Fréchet theorem
• 3. Sesquilinear forms and adjoints
• 8. Hilbert tensor products
• Adjoining an identity
• A C\*-norm on the unitization
• The resolvent; the spectrum is compact
• 4. Homomorphisms: contractivity, isometry and automatic continuity
• 8. The positive cone and the order
• 11. Approximate identities
• 15. Closed ideals and quotients
• 3. The Banach–Alaoglu theorem
• 6. The Krein–Milman theorem
• 15. Closed ideals and quotients
• Q25-GNS-ESSENTIAL: Arbitrary Hilbert spaces; zero and nondegenerate summands; arbitrary bounded approximate-identity nets
• Q25-GNS-NULL-SPACE: Nonunital star algebras and arbitrary positive functionals; no boundedness presumed
• Q25-GNS-COMPRESSED-BOUND: C* compressed functional estimate and bounded quotient multiplication only; broader old Banach-star theorem explicitly outside this replacement
• Q25-GNS-STATE-BOUNDS: Arbitrary nonunital C* algebra, arbitrary positive functional and positive contractive approximate-identity net
• Q25-GNS-CYCLIC: Arbitrary nonunital C* algebra and bounded state; cyclic unit vector; nondegenerate Hilbert representation
• Q25-GNS-FAITHFUL: Arbitrary nonunital C* algebra; faithful direct sum of cyclic nondegenerate representations

Locally compact cutoff functions

The Stone–Weierstrass theorem for functions vanishing at infinity: Proposition 2.1(3) and Corollary 5.2, for density and compact-set cutoff functions.