{ "schema": "public-mathematical-scope/v1", "targets": [ { "target_id": "TT1-III.6.29", "lesson": "src/abelian-semicontinuity-and-multiplier-spectra.md", "scope": "Finite-Radon increasing-net identity on arbitrary locally compact Hausdorff spectrum; isometric lower/upper function models, norm closedness and open/closed indicators", "proof_locator": "Lemma 1.1 / Theorem 2.1 / Corollary 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AC/Lemma-1.1", "OA-FOUND-REMAINDER/AC/Theorem-2.1", "OA-FOUND-REMAINDER/AC/Corollary-2.2" ] }, { "target_id": "TT1-III.6.30", "lesson": "src/abelian-semicontinuity-and-multiplier-spectra.md", "scope": "Complete multiplier identification with C_b; Stone-Cech compactification by the stronger exact CF Exercise2.4 full proof", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AC/Theorem-3.1" ] }, { "target_id": "TT1-III.6.1", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Finite lower semicontinuous convex function is supremum of continuous ambient affine minorants", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Theorem-1.1" ] }, { "target_id": "TT1-III.6.2", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Lower semicontinuous affine function has an increasing net of continuous ambient affine strict minorants", "proof_locator": "Theorem 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Theorem-1.2" ] }, { "target_id": "TT1-III.6.3", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Continuous affine function uniformly approximated by an increasing sequence, without metrizability", "proof_locator": "Corollary 1.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Corollary-1.3" ] }, { "target_id": "TT1-III.6.4", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Minimum attained at an extreme point using a compact minimizing face", "proof_locator": "Proposition 1.4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Proposition-1.4" ] }, { "target_id": "TT1-III.6.5", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Bounded affine extension approximated from below along a split face with lower semicontinuous weight", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Theorem-2.1" ] }, { "target_id": "TT1-III.6.6", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Singleton-complement normalization with continuous affine term vanishing there and weight correction", "proof_locator": "Corollary 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Corollary-2.2" ] }, { "target_id": "TT1-III.6.7", "lesson": "src/affine-approximation-and-quasi-state-spaces.md", "scope": "Complete continuous and bounded affine quasi-state models for A and A**, including nonunital states/zero split", "proof_locator": "Theorems 3.1–3.2 / Section 4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AF/Theorem-3.1", "OA-FOUND-REMAINDER/AF/Theorem-3.2", "OA-FOUND-REMAINDER/AF/States-and-zero-split", "OA-FOUND-REMAINDER/AF/Example-4.1" ] }, { "target_id": "TT1-III.6.35", "lesson": "src/atomic-representations-and-measurable-lifts.md", "scope": "Universal atomic representation as direct sum of all pure-state GNS representations", "proof_locator": "Section 1 / formula (1.1)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AT/Universal-atomic-representation" ] }, { "target_id": "TT1-III.6.36", "lesson": "src/atomic-representations-and-measurable-lifts.md", "scope": "Pure states and minimal normal supports in both directions; atomic carrier; full central-support/quasi-equivalence characterization, including weak image of degenerate representation", "proof_locator": "Lemma 1.1 / Theorem 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AT/Lemma-1.1", "OA-FOUND-REMAINDER/AT/Theorem-1.2" ] }, { "target_id": "TT1-III.6.37", "lesson": "src/atomic-representations-and-measurable-lifts.md", "scope": "Order detection by pure states, compact quasi-state extreme minimizer, and isometry on the measurable self-adjoint space", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AT/Theorem-2.1" ] }, { "target_id": "TT1-III.6.38", "lesson": "src/atomic-representations-and-measurable-lifts.md", "scope": "Measurable majorant of atomic carrier majorizes identity; proper atomic carrier excluded, with explicit arbitrary-net example", "proof_locator": "Corollary 2.2 / Exercise 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AT/Corollary-2.2", "OA-FOUND-REMAINDER/AT/Exercise-4.2" ] }, { "target_id": "TT1-III.6.39", "lesson": "src/atomic-representations-and-measurable-lifts.md", "scope": "Sigma-finite weak image has measurable self-adjoint lifts preserving norm, positivity and projections; exact stronger KD projection lemma/dyadic expansion reused; non-sigma-finite counterexample", "proof_locator": "Theorem 3.1 / Exercise 4.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AT/Theorem-3.1", "OA-FOUND-REMAINDER/AT/Exercise-4.3" ] }, { "target_id": "TT1-III.Notes", "lesson": "src/atomic-representations-and-measurable-lifts.md", "scope": "Complete assigned chapter Notes synthesis: existing WA/PD/ABA results and historical references reused, new semicontinuity/multiplier attributions and approximation/Borel motivation supplied", "proof_locator": "Section 5 / References", "explicit_result_ids": [ "OA-FOUND-REMAINDER/AT/Historical-setting" ] }, { "target_id": "TT1-I.10.1(i)", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Commutative unit-ball extreme points; nonunital case", "proof_locator": "Lemma 1.1(1)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Lemma-1.1" ] }, { "target_id": "TT1-I.10.1(ii)", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Positive-contraction extremes; multiplicative perturbation; stronger noncommutative positive ball", "proof_locator": "Lemma 1.1(2–3), Proposition 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Lemma-1.1", "OA-FOUND-REMAINDER/FD/Proposition-1.2" ] }, { "target_id": "TT1-I.10.2(i)", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Extreme point existence iff unital; approximate identity limit", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-2.1" ] }, { "target_id": "TT1-I.10.2(ii)", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Full defect-corner characterization; both equivalent orientations", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-2.1" ] }, { "target_id": "TT1-I.11.1", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Unit and central units of all ideals", "proof_locator": "Lemma 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Lemma-3.1" ] }, { "target_id": "TT1-I.11.2", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Matrix block classification, existence and uniqueness", "proof_locator": "Theorem 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-3.2" ] }, { "target_id": "TT1-I.11.3", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Matrix unit definition; sum identity specified", "proof_locator": "Section 3, equation (3.1)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Matrix-unit-conventions" ] }, { "target_id": "TT1-I.11.4", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Minimality defined by pAp=Cp; finite-dimensional lattice equivalence", "proof_locator": "Conventions; Section 3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Minimal-projection-conventions", "OA-FOUND-REMAINDER/FD/Theorem-3.2" ] }, { "target_id": "TT1-I.11.5", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Arbitrary-cardinal multiplicity, independence of minimal projection, dimension and equivalence", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-4.1" ] }, { "target_id": "TT1-I.11.6", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Irreducibility iff multiplicity one; uniqueness", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-4.1" ] }, { "target_id": "TT1-I.11.7", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Every cardinal multiplicity, including zero", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-4.1" ] }, { "target_id": "TT1-I.11.8", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Commutant and reverse multiplicity; arbitrary Hilbert-space strengthening", "proof_locator": "Proposition 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Proposition-4.2" ] }, { "target_id": "TT1-I.11.9", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "All blocks; zero multiplicities; finite commutants; degenerate null summand separated", "proof_locator": "Corollary 4.3 and following paragraph", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Corollary-4.3" ] }, { "target_id": "TT1-I.11.Ex1", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Finite-dimensional MASA iff finite-dimensional algebra, initially nonunital", "proof_locator": "Theorem 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-5.1" ] }, { "target_id": "TT1-I.11.Ex2", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Reflexive iff finite-dimensional; explicit closed c0 construction", "proof_locator": "Theorem 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-5.2" ] }, { "target_id": "TT1-I.11.Ex3", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "Simple unital algebra with minimal projection; repair algebraic-ideal closure argument", "proof_locator": "Theorem 5.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Theorem-5.3" ] }, { "target_id": "TT1-I.Notes", "lesson": "src/extreme-points-and-matrix-blocks.md", "scope": "All historical/pedagogical topics and seventeen numbered bibliography references traced; quotient choice commentary qualified; asymmetric Riesz theorem located at actual statement; positivity pointer, Gelfand–Naimark pages and Sinclair date corrected with distinct metadata-reading scope.", "proof_locator": "Section 7 and References", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FD/Historical-notes" ] }, { "target_id": "TT1-I.Ex1(a)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Polar phase and absolute value remain Fredholm", "proof_locator": "Corollary 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Corollary-1.2" ] }, { "target_id": "TT1-I.Ex1(b)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Closed range of self-adjoint Fredholm operators", "proof_locator": "Theorem 1.1; Corollary 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Theorem-1.1", "OA-FOUND-REMAINDER/FH/Corollary-1.2" ] }, { "target_id": "TT1-I.Ex1(c)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Full finite-kernel/cokernel/closed-range characterization", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Theorem-1.1" ] }, { "target_id": "TT1-I.Ex1(d)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Index homomorphism; exact-sequence proof; source sign retained", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Theorem-2.1" ] }, { "target_id": "TT1-I.Ex1(e)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Close projections have equal dimensions, arbitrary cardinals", "proof_locator": "Lemma 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Lemma-3.1" ] }, { "target_id": "TT1-I.Ex1(f)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Source false nullity-continuity and polar-continuity replaced by upper semicontinuity, local index constancy and rank-one counterexample", "proof_locator": "Theorem 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Theorem-3.2" ] }, { "target_id": "TT1-I.Ex1(g)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Compact perturbation invariance", "proof_locator": "Corollary 3.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Corollary-3.3" ] }, { "target_id": "TT1-I.Ex1(h)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Polar phase has same index", "proof_locator": "Corollary 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Corollary-1.2" ] }, { "target_id": "TT1-I.Ex1(i)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Partial isometry is unitary plus compact iff zero index", "proof_locator": "Proposition 3.4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Proposition-3.4" ] }, { "target_id": "TT1-I.Ex1(j)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Identity component of Calkin invertibles iff zero index", "proof_locator": "Theorem 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Theorem-4.2" ] }, { "target_id": "TT1-I.Ex1(k)", "lesson": "src/fredholm-operators-and-stable-index.md", "scope": "Calkin component group is Z in full assigned separable infinite-dimensional case", "proof_locator": "Theorem 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FH/Theorem-4.2" ] }, { "target_id": "TT1-I.Ex2(a)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Continuous invertible polar unitary", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-4.1" ] }, { "target_id": "TT1-I.Ex2(b)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Invertibles equal unitaries times identity component", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-4.1" ] }, { "target_id": "TT1-I.Ex2(c)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Unitary identity component equals intersection", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-4.1" ] }, { "target_id": "TT1-I.Ex2(d)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Unitary and invertible component groups isomorphic", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-4.1" ] }, { "target_id": "TT1-I.Ex3(a)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Continuous argument lift on real line", "proof_locator": "Lemma 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Lemma-5.1" ] }, { "target_id": "TT1-I.Ex3(b)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Circle pullback lift", "proof_locator": "Lemma 5.1 / equation (5.1)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Lemma-5.1" ] }, { "target_id": "TT1-I.Ex3(c)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Integer degree; homomorphism and surjectivity", "proof_locator": "Theorem 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-5.2" ] }, { "target_id": "TT1-I.Ex3(d)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Kernel exactly unitary principal component and component group Z", "proof_locator": "Theorem 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-5.2" ] }, { "target_id": "TT1-I.Ex4(a)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Every norm-continuous Banach-algebra one-parameter group has bounded generator", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-2.1" ] }, { "target_id": "TT1-I.Ex4(b)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Identity component generated by exponentials", "proof_locator": "Recall 1.1; imported BN Proposition 7.1(2),(5)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Recall-1.1" ] }, { "target_id": "TT1-I.Ex4(c)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Norm Trotter formula for arbitrary Banach algebra", "proof_locator": "Theorem 3.1, equation (3.1)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-3.1" ] }, { "target_id": "TT1-I.Ex4(d)", "lesson": "src/invertible-components-and-exponential-laws.md", "scope": "Norm commutator formula and sign with n-squared power", "proof_locator": "Theorem 3.1, equation (3.2)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EXP/Theorem-3.1" ] }, { "target_id": "TT1-III.6.8", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "All four equivalences; explicit directed, bounded A approximants to every positive scalar shift; common lower bound", "proof_locator": "Theorem 2.1 / Remark 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Theorem-2.1" ] }, { "target_id": "TT1-III.6.9", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "Every positive scalar shift of positive lower semicontinuous element has bounded positive increasing approximants", "proof_locator": "Proposition 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Proposition-3.1" ] }, { "target_id": "TT1-III.6.10", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "Rational stability for small parameter in self-adjoint cone and all positive parameters in positive cone", "proof_locator": "Corollary 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Corollary-3.2" ] }, { "target_id": "TT1-III.6.11", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "Exact unitized upper cone equals scalar plus norm-closed original upper cone iff bounded lsc affine extension from states", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Theorem-4.1" ] }, { "target_id": "TT1-III.6.12", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "Norm-closed differences agree before and after adjoining the bidual identity", "proof_locator": "Formulas (4.1)–(4.2)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Theorem-4.1", "OA-FOUND-REMAINDER/MS/Identity-adjoined-difference-space" ] }, { "target_id": "TT1-III.6.13", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "Both one-sided weak-to-strong resolvent estimates, including unbounded net families; rational operator monotonicity and bounded strong calculus", "proof_locator": "Lemma 5.1 / Section 1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Lemma-5.1" ] }, { "target_id": "TT1-III.6.14", "lesson": "src/monotone-approximation-and-semicontinuous-operators.md", "scope": "Norm-closed Jordan algebra and continuous one-variable functional calculus", "proof_locator": "Theorem 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MS/Theorem-6.1" ] }, { "target_id": "TT1-III.6.22", "lesson": "src/multipliers-and-essential-extensions.md", "scope": "All three multiplier definitions; norm-closed unital C*-algebra and ideal property", "proof_locator": "Section 1 / Proposition 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MU/Proposition-1.1" ] }, { "target_id": "TT1-III.6.23", "lesson": "src/multipliers-and-essential-extensions.md", "scope": "Essential/thick ideal definition, equivalent annihilator tests and essentiality of A in its multiplier algebra", "proof_locator": "Lemma 1.2 / Corollary 1.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MU/Lemma-1.2", "OA-FOUND-REMAINDER/MU/Corollary-1.3" ] }, { "target_id": "TT1-III.6.24", "lesson": "src/multipliers-and-essential-extensions.md", "scope": "Exact intersection of bounded upper and lower monotone-limit classes after adjoining the bidual identity; no added norm closures", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MU/Theorem-2.1" ] }, { "target_id": "TT1-III.6.25", "lesson": "src/multipliers-and-essential-extensions.md", "scope": "Exact complete WA Proposition5.7 ideal-representation extension reused; multiplier range, faithfulness from essentiality, uniqueness and unitality", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/MU/Theorem-3.1" ] }, { "target_id": "TT1-III.6.15", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "All four state-space lower semicontinuity equivalences; nonmonotone one-sided resolvent supremum and unitized upper-cone transforms", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Theorem-1.1" ] }, { "target_id": "TT1-III.6.16", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "Both norm-closed semicontinuity cones closed under bounded increasing nets", "proof_locator": "Corollary 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Corollary-1.2" ] }, { "target_id": "TT1-III.6.17", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "Both directions of positive scalar shift versus bounded positive transform", "proof_locator": "Proposition 1.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Proposition-1.3" ] }, { "target_id": "TT1-III.6.18", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "State-semicontinuous projection has exact positive increasing A approximants; bounded-ball proof of annihilator weak-star closedness", "proof_locator": "Lemmas 2.1–2.2 / Theorem 2.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Lemma-2.1", "OA-FOUND-REMAINDER/OP/Lemma-2.2", "OA-FOUND-REMAINDER/OP/Theorem-2.3" ] }, { "target_id": "TT1-III.6.19", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "Open and closed projection definitions relative to the original algebra", "proof_locator": "Section 3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Open-and-closed-projection-conventions" ] }, { "target_id": "TT1-III.6.20", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "All source one-sided ideal/annihilator/increasing-net equivalences, plus state semicontinuity, at arbitrary generality", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Theorem-3.1" ] }, { "target_id": "TT1-III.6.21", "lesson": "src/open-projections-and-closed-one-sided-ideals.md", "scope": "Ideal bidual identification and norm closure of central compression of the global unitized upper cone; closure retained explicitly", "proof_locator": "Proposition 4.1 / Exercise 5.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OP/Proposition-4.1", "OA-FOUND-REMAINDER/OP/Exercise-5.3" ] }, { "target_id": "TT1-III.6.26", "lesson": "src/split-faces-and-semicontinuous-quotient-lifts.md", "scope": "Bounded affine lower/upper semicontinuous extension from a closed split face, with positivity and compact full-space upper strip", "proof_locator": "Lemma 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/QD/Lemma-1.1" ] }, { "target_id": "TT1-III.6.27", "lesson": "src/split-faces-and-semicontinuous-quotient-lifts.md", "scope": "Closed quotient quasi-state and state split faces, complementary ideal states, endpoint cases and both correct central summands", "proof_locator": "Proposition 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/QD/Proposition-2.1" ] }, { "target_id": "TT1-III.6.28", "lesson": "src/split-faces-and-semicontinuous-quotient-lifts.md", "scope": "All three onto cone equalities; explicit norm-preserving central lifts for both norm-closed classes and positive lifts for positive elements", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/QD/Theorem-3.1" ] }, { "target_id": "TT1-IV.4.3", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Faithful extension of every norm to genuine unitizations", "proof_locator": "Lemma 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Lemma-5.1" ] }, { "target_id": "TT1-IV.4.11", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Pure restriction factorization across relative commutant", "proof_locator": "Lemma 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Lemma-2.2" ] }, { "target_id": "TT1-IV.4.12", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Remark disposition: commutative Gelfand theorem is an existing CF import, no alternative-proof duplication", "proof_locator": "Theorem 1.3 / Theorem 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Theorem-1.3", "OA-FOUND-REMAINDER/SI/Theorem-4.2" ] }, { "target_id": "TT1-IV.4.13", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Weak closure of product representations and tensor irreducibility", "proof_locator": "Proposition 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Proposition-4.1" ] }, { "target_id": "TT1-IV.4.14", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Commutativity, pure product states and Banach injective norm equivalence; nonunital extensions", "proof_locator": "Theorem 4.2 / Section 5", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Theorem-4.2" ] }, { "target_id": "TT1-IV.4.15", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Closed unitary-invariant pure sets and ideals, both inverse directions", "proof_locator": "Lemma 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Lemma-2.1" ] }, { "target_id": "TT1-IV.4.16", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Unitary invariance of product-pure continuity locus", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Theorem-3.1" ] }, { "target_id": "TT1-IV.4.17", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Relative closedness of continuity locus", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Theorem-3.1" ] }, { "target_id": "TT1-IV.4.18", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Uniqueness when one factor is commutative; independent CP partition proof", "proof_locator": "Theorem 1.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Theorem-1.3" ] }, { "target_id": "TT1-IV.4.19", "lesson": "src/states-ideals-and-the-smallest-tensor-norm.md", "scope": "Full minimality, both factors possibly nonunital", "proof_locator": "Theorem 3.1 / Section 5", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SI/Theorem-3.1" ] }, { "target_id": "TT1-IV.4.20", "lesson": "src/tensor-independence-and-ideals.md", "scope": "Factor and commutant algebraic multiplication injectivity", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CI/Theorem-1.1" ] }, { "target_id": "TT1-IV.4.21", "lesson": "src/tensor-independence-and-ideals.md", "scope": "Simplicity, including nonunital simple factors", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CI/Theorem-2.1" ] }, { "target_id": "TT1-IV.4.Ex2", "lesson": "src/tensor-independence-and-ideals.md", "scope": "Commuting algebras, one commutative, no zero elementary product gives spatial identification", "proof_locator": "Corollary 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CI/Corollary-3.2" ] }, { "target_id": "TT1-IV.4.Ex3a-c", "lesson": "src/tensor-independence-and-ideals.md", "scope": "Product pure states detect ideal; invariant open rectangle; product ideal containment", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CI/Theorem-3.1" ] }, { "target_id": "TT1-IV.4.Ex4.tensor_norm_nonuniqueness", "lesson": "src/tensor-independence-and-ideals.md", "scope": "Explicit tensor norm 4 versus 2 sqrt(3); no reliance on reduced free-group simplicity", "proof_locator": "Proposition 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CI/Proposition-4.1" ] }, { "target_id": "TT1-IV.4.1", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Nondegenerate representations; unique restrictions; arbitrary approximate identities; bounds", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Theorem-1.1" ] }, { "target_id": "TT1-IV.4.2", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Restriction terminology retained as recovered factor actions", "proof_locator": "Section 1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Theorem-1.1" ] }, { "target_id": "TT1-IV.4.4", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Real self-adjoint tensor span and algebraic order-unit bound", "proof_locator": "Lemma 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Lemma-5.1" ] }, { "target_id": "TT1-IV.4.5", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Maximal norm, cross property, finiteness, completion", "proof_locator": "Section 2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Proposition-2.1", "OA-FOUND-REMAINDER/TN/Proposition-2.2" ] }, { "target_id": "TT1-IV.4.6", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Algebraic state iff CP dual map; specified pairing; normalization", "proof_locator": "Proposition 5.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Proposition-5.3" ] }, { "target_id": "TT1-IV.4.7", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Commuting homomorphism universal property; correct nonunital product range", "proof_locator": "Proposition 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Proposition-2.2" ] }, { "target_id": "TT1-IV.4.8", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Minimal norm and faithful spatial computation", "proof_locator": "Section 2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Proposition-2.1", "OA-FOUND-REMAINDER/TN/Theorem-3.1" ] }, { "target_id": "TT1-IV.4.9", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Product GNS; norm formula; faithful spatial representations", "proof_locator": "Section 3 / Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Theorem-3.1", "OA-FOUND-REMAINDER/TN/Product-GNS" ] }, { "target_id": "TT1-IV.4.10", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Equal dual norms of finite sums of product functionals", "proof_locator": "Proposition 4.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Proposition-4.3", "OA-FOUND-REMAINDER/TN/Product-functional-norm" ] }, { "target_id": "TT1-IV.4.22", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Minimal functoriality and injectivity for injective factor maps", "proof_locator": "Corollary 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Corollary-3.2" ] }, { "target_id": "TT1-IV.4.23", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "CP minimal tensor and commuting CP maximal product; bounded nonunital maps", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Theorem-4.1" ] }, { "target_id": "TT1-IV.4.24", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Commuting Schur product and block-index scalar compression", "proof_locator": "Lemma 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Lemma-6.1" ] }, { "target_id": "TT1-IV.4.25", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Positive slices; norm; bimodularity without units", "proof_locator": "Corollary 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Corollary-4.2" ] }, { "target_id": "TT1-IV.4.Ex1a-b", "lesson": "src/tensor-norms-and-independent-systems.md", "scope": "Exact partial transpose norm n and unbounded infinite compact-operator tensor transpose", "proof_locator": "Exercise 7.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/TN/Exercise-7.2" ] }, { "target_id": "TT1-III.6.31", "lesson": "src/universal-measurability-and-strong-sequences.md", "scope": "Universal measurability defined with original bounded monotone classes and arbitrarily small state gaps", "proof_locator": "Section 1 / formula (1.1)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/UM/State-gap-definition" ] }, { "target_id": "TT1-III.6.32", "lesson": "src/universal-measurability-and-strong-sequences.md", "scope": "Exact nested measurable squeezing criterion, norm-closed real vector space, affine invariance and Jordan-class inclusion", "proof_locator": "Lemma 1.1 / Proposition 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/UM/Lemma-1.1", "OA-FOUND-REMAINDER/UM/Proposition-1.2" ] }, { "target_id": "TT1-III.6.33", "lesson": "src/universal-measurability-and-strong-sequences.md", "scope": "Intrinsic normal-state quadratic criterion; individual-vector criterion in universal representation; equivalent strong closure on finite vector sets in every faithful normal realization, with full bounded-transform converse", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/UM/Theorem-2.1" ] }, { "target_id": "TT1-III.6.34", "lesson": "src/universal-measurability-and-strong-sequences.md", "scope": "Monotone then arbitrary sequential strong closure; state-bounded unbounded sums rationally compressed; corrected initial-error estimate", "proof_locator": "Lemma 3.1 / Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/UM/Lemma-3.1", "OA-FOUND-REMAINDER/UM/Theorem-4.1" ] }, { "target_id": "TT1-V.4.1", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Central comparison and the three-quarter corner-width contraction", "proof_locator": "Lemma 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Lemma-1.1" ] }, { "target_id": "TT1-V.4.2", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Simultaneous central partition refinement and corner unitaries", "proof_locator": "Corollary 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Corollary-1.2" ] }, { "target_id": "TT1-V.4.3", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Self-adjoint finite average arbitrarily close to centre", "proof_locator": "Corollary 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Corollary-1.2" ] }, { "target_id": "TT1-V.4.4", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "One finite average for every element of a finite family", "proof_locator": "Lemma 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Lemma-2.1" ] }, { "target_id": "TT1-V.4.5", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Successive finite averages converge in norm to simultaneous central limits", "proof_locator": "Theorem 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Theorem-2.2" ] }, { "target_id": "TT1-V.4.6(i–ii)", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Both norm/ultraweak central hulls; singleton for every element iff finite; centre-valued trace reuse", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Theorem-3.1" ] }, { "target_id": "TT1-V.4.6(iii)", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Nonzero central hull point for every nonzero element of sigma-finite type III algebra", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Theorem-4.1" ] }, { "target_id": "TT1-V.4.7", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Centre-valued trace of closed ideal equals its central intersection; exact TY simplicity reuse", "proof_locator": "Corollary 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Corollary-3.2" ] }, { "target_id": "TT1-V.4.8", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Full maximal-ideal correspondence and precise algebraic-ideal closure statement; page-image confirms closures lost in extraction", "proof_locator": "Lemmas 5.1–5.2 / Theorem 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Lemma-5.1", "OA-FOUND-REMAINDER/CA/Lemma-5.2", "OA-FOUND-REMAINDER/CA/Theorem-6.1" ] }, { "target_id": "TT1-V.4.9", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Every factor has exactly one maximal proper ideal", "proof_locator": "Theorem 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Theorem-6.1" ] }, { "target_id": "TT1-V.4.10", "lesson": "src/central-averaging-and-maximal-ideals.md", "scope": "Finite maximal ideal characterized by central character of T(x*x)", "proof_locator": "Corollary 6.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CA/Corollary-6.2" ] }, { "target_id": "TT1-V.5.2", "lesson": "src/finite-maximal-quotients.md", "scope": "Arbitrary finite M; central patches and unit-ball vector closedness; maximal quotient finite factor; type-I product/type-II quotient example", "proof_locator": "Theorem 3.1 / Section 4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FQ/Lemma-1.1", "OA-FOUND-REMAINDER/FQ/Lemma-2.1", "OA-FOUND-REMAINDER/FQ/Theorem-3.1", "OA-FOUND-REMAINDER/FQ/Type-I-product-and-type-II-quotient" ] }, { "target_id": "TT1-IV.4.Ex4.simplicity_of_left_and_right_reduced_free_group_algebras", "lesson": "src/free-group-averaging-and-the-compact-ideal.md", "scope": "Explicit word partition, full norm-averaging estimate, faithful canonical trace, simplicity and unique trace of both reduced factors", "proof_locator": "Lemmas 1.1–1.2 / Proposition 1.3 / Theorem 1.4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FG/Lemma-1.1", "OA-FOUND-REMAINDER/FG/Lemma-1.2", "OA-FOUND-REMAINDER/FG/Proposition-1.3", "OA-FOUND-REMAINDER/FG/Theorem-1.4" ] }, { "target_id": "TT1-IV.4.Ex4.compact_operators_inside_joint_algebra_and_unique_nontrivial_ideal", "lesson": "src/free-group-averaging-and-the-compact-ideal.md", "scope": "Full cyclic-centralizer proof and coset approximation transfer spectral gap to conjugation; rank-one projection and all compacts; complete ideal lattice", "proof_locator": "Lemmas 2.1–2.2 / Proposition 2.3 / Corollary 2.4 / Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FG/Lemma-2.1", "OA-FOUND-REMAINDER/FG/Lemma-2.2", "OA-FOUND-REMAINDER/FG/Proposition-2.3", "OA-FOUND-REMAINDER/FG/Corollary-2.4", "OA-FOUND-REMAINDER/FG/Theorem-4.1" ] }, { "target_id": "TT1-IV.4.Ex4.canonical_quotient_identification_with_minimal_tensor_product", "lesson": "src/free-group-averaging-and-the-compact-ideal.md", "scope": "Explicit averaged word-cut isometry, cancellation bound and orthogonal columns prove compact intertwining and spatial quotient continuity; simple product gives canonical isomorphism", "proof_locator": "Lemmas 3.1 / Proposition 3.2 / Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FG/Lemma-3.1", "OA-FOUND-REMAINDER/FG/Proposition-3.2", "OA-FOUND-REMAINDER/FG/Theorem-4.1" ] }, { "target_id": "TT1-V.5.1(i).finite_type_II_part", "lesson": "src/finite-type-ii-and-separable-representations.md", "scope": "Sigma-finite finite type II algebra with arbitrary centre; both Radon-measure cases, balanced chosen abelian algebra, full no-finite-type-I conclusion", "proof_locator": "Theorem 5.1 / Corollary 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/FI/Lemma-1.1", "OA-FOUND-REMAINDER/FI/Lemma-2.1", "OA-FOUND-REMAINDER/FI/Lemma-2.2", "OA-FOUND-REMAINDER/FI/Lemma-4.1", "OA-FOUND-REMAINDER/FI/Lemma-4.2", "OA-FOUND-REMAINDER/FI/Theorem-5.1", "OA-FOUND-REMAINDER/FI/Corollary-5.2" ] }, { "target_id": "TT1-IV.8.26", "lesson": "src/proper-infiniteness-and-automatic-normality.md", "scope": "Full nonseparable diagonal-commutant field obstructed on diffuse part by automatic normality; exact atomic boundary proved", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/PI/Theorem-4.1" ] }, { "target_id": "TT1-V.5.1(ii)", "lesson": "src/proper-infiniteness-and-automatic-normality.md", "scope": "Sigma-finite properly infinite domain; generated range sigma-finite", "proof_locator": "Theorem 2.1 / Lemma 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/PI/Lemma-1.1", "OA-FOUND-REMAINDER/PI/Corollary-1.2", "OA-FOUND-REMAINDER/PI/Theorem-2.1" ] }, { "target_id": "TT1-V.5.1(i).properly_infinite_part", "lesson": "src/proper-infiniteness-and-automatic-normality.md", "scope": "Separable target Hilbert space; properly infinite part", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/PI/Theorem-2.1" ] }, { "target_id": "TT1-V.5.Ex1", "lesson": "src/proper-infiniteness-and-automatic-normality.md", "scope": "Calkin algebra has no nonzero representation on separable Hilbert space", "proof_locator": "Corollary 3.1 / Exercise 6.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/PI/Corollary-3.1", "OA-FOUND-REMAINDER/PI/Exercise-6.2" ] }, { "target_id": "TT1-V.5.Ex2", "lesson": "src/proper-infiniteness-and-automatic-normality.md", "scope": "Surjective homomorphisms from properly infinite separable-predual algebras are normal; arbitrary target", "proof_locator": "Lemma 5.1 / Theorem 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/PI/Lemma-5.1", "OA-FOUND-REMAINDER/PI/Theorem-5.2" ] }, { "target_id": "TT1-V.4.Ex1(a)", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Invariant normal positive functional; compact GNS image and exact minimizing fixed vector", "proof_locator": "Proposition 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Proposition-1.1" ] }, { "target_id": "TT1-V.4.Ex1(b)", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Sum-GNS reducing embedding and exact finite intersection of minimizing sets", "proof_locator": "Proposition 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Proposition-1.2" ] }, { "target_id": "TT1-V.4.Ex1(c)", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Separating family, arbitrary group; compact finite intersection and unique fixed orbit-hull point", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Theorem-2.1" ] }, { "target_id": "TT1-V.4.Ex1(d)", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Unique faithful G-invariant normal norm-one projection; tuple averaging, linearity, bimodule property, normality", "proof_locator": "Theorems 2.1–2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Theorem-2.1", "OA-FOUND-REMAINDER/IS/Theorem-2.2" ] }, { "target_id": "TT1-V.4.Ex1(e)", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "All predual orbits relatively weakly compact; complete domination, arbitrary-group fixed-point and full closed-convex-hull compactness proofs supplied", "proof_locator": "Lemmas 3.1–3.2 / Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Lemma-3.1", "OA-FOUND-REMAINDER/IS/Lemma-3.2", "OA-FOUND-REMAINDER/IS/Theorem-4.1" ] }, { "target_id": "TT1-V.4.Ex1(f)", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Exact topology from V.2.Ex6: weak-star closure in bounded maps remains normal; page-image verified", "proof_locator": "Theorem 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Bounded-map-duality-and-compact-closure", "OA-FOUND-REMAINDER/IS/Theorem-5.1" ] }, { "target_id": "TT1-V.4.Ex2", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Factor finite iff full automorphism group closure consists of normal maps", "proof_locator": "Corollary 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Corollary-5.2" ] }, { "target_id": "TT1-V.4.Ex3", "lesson": "src/invariant-states-and-ergodic-projections.md", "scope": "Inner action satisfies large-group hypothesis via tracial GNS and unitary Kaplansky bridge; exact IR Theorems 24.1(b) and 23.5 reused", "proof_locator": "Proposition 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IS/Proposition-6.1" ] }, { "target_id": "TT1-IV.8.33", "lesson": "src/normal-functionals-across-fibres.md", "scope": "Measurable varying-fibre normal functional fields and integrable norm", "proof_locator": "Section 1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NF/Measurable-norm-detecting-sequence", "OA-FOUND-REMAINDER/NF/Lemma-1.1" ] }, { "target_id": "TT1-IV.8.34", "lesson": "src/normal-functionals-across-fibres.md", "scope": "Isometric onto varying predual integral; normality approximation on arbitrary sigma-finite base", "proof_locator": "Theorem 4.1 / Sections 2–4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NF/Proposition-2.1", "OA-FOUND-REMAINDER/NF/Lemma-3.1", "OA-FOUND-REMAINDER/NF/Theorem-3.2", "OA-FOUND-REMAINDER/NF/Theorem-4.1", "OA-FOUND-REMAINDER/NF/Corollary-4.2", "OA-FOUND-REMAINDER/NF/Corollary-4.3" ] }, { "target_id": "TT1-V.6.6", "lesson": "src/borel-types-and-fibre-types.md", "scope": "Joint finite-corner relation Borel; type I and finite loci Borel; semifinite and complement of pure III analytic; factor consequences", "proof_locator": "Proposition 1.2 / Theorems 2.1, 2.3 / Corollary 2.4", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BT/Proposition-1.2", "OA-FOUND-REMAINDER/BT/Theorem-2.1", "OA-FOUND-REMAINDER/BT/Theorem-2.3", "OA-FOUND-REMAINDER/BT/Corollary-2.4" ] }, { "target_id": "TT1-V.6.7", "lesson": "src/borel-types-and-fibre-types.md", "scope": "Finite projection transfer at arbitrary algebra-fibre generality; all six properties of factor integrals iff almost every fibre; completed-measure witnesses and normal trace integration", "proof_locator": "Lemma 3.1 / Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BT/Lemma-3.1", "OA-FOUND-REMAINDER/BT/Theorem-4.1", "OA-FOUND-REMAINDER/BT/Corollary-4.3" ] }, { "target_id": "TT1-V.6.2", "lesson": "src/borel-supports-and-isomorphism-classes.md", "scope": "Joint Borel smallest M-projection majorizing arbitrary ambient p", "proof_locator": "Proposition 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BS/Proposition-2.1" ] }, { "target_id": "TT1-V.6.3", "lesson": "src/borel-supports-and-isomorphism-classes.md", "scope": "Trace-class support and support of normal restriction jointly Borel", "proof_locator": "Proposition 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BS/Proposition-3.1" ] }, { "target_id": "TT1-V.6.4", "lesson": "src/borel-supports-and-isomorphism-classes.md", "scope": "Complete closed-set selector proof and both closed-subgroup coset transversals", "proof_locator": "Lemma 4.1 / Corollary 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BS/Lemma-4.1", "OA-FOUND-REMAINDER/BS/Corollary-4.2" ] }, { "target_id": "TT1-V.6.5", "lesson": "src/borel-supports-and-isomorphism-classes.md", "scope": "Fixed unitary class Borel; fixed abstract isomorphism class Borel by normal faithful amplification", "proof_locator": "Theorems 5.1–5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BS/Theorem-5.1", "OA-FOUND-REMAINDER/BS/Theorem-5.2" ] }, { "target_id": "TT1-V.Notes", "lesson": "src/borel-supports-and-isomorphism-classes.md", "scope": "Historical topics mapped to exact lessons; cited bibliography identities compared, two source cross-reference issues recorded; later Nielsen refinement remains historical, not an admitted extra proof", "proof_locator": "Historical setting / References", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BS/Historical-setting" ] }, { "target_id": "TT1-IV.8.28", "lesson": "src/measurable-equivalence-and-constant-fibres.md", "scope": "Measurable choices of spatial equivalences for algebra and representation fields", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/ME/Theorem-2.1" ] }, { "target_id": "TT1-IV.8.29", "lesson": "src/measurable-equivalence-and-constant-fibres.md", "scope": "Measurable abstract normal isomorphisms: target range, full central carrier, compressed commutant and measurable inverse", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/ME/Theorem-3.1" ] }, { "target_id": "TT1-IV.8.30", "lesson": "src/measurable-equivalence-and-constant-fibres.md", "scope": "Constant abstract fibre type gives diagonal-preserving L-infinity tensor product", "proof_locator": "Corollary 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/ME/Corollary-4.1" ] }, { "target_id": "TT1-V.7.11", "lesson": "src/expected-masas-and-factor-types.md", "scope": "Arbitrary expected MASA: compact commuting Cesaro averages, uniqueness of normal expectation, semifinite trace restriction and trace factorization", "proof_locator": "Lemmas 1.1 / Theorem 1.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EM/Lemma-1.1", "OA-FOUND-REMAINDER/EM/Theorem-1.2" ] }, { "target_id": "TT1-V.7.12", "lesson": "src/expected-masas-and-factor-types.md", "scope": "Invariant semifinite trace correspondence and uniqueness; full four-type criterion at arbitrary coefficient-algebra generality for countably infinite free ergodic actions", "proof_locator": "Proposition 2.1 / Lemma 3.1 / Theorem 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EM/Proposition-2.1", "OA-FOUND-REMAINDER/EM/Lemma-3.1", "OA-FOUND-REMAINDER/EM/Theorem-3.2" ] }, { "target_id": "TT1-V.7.13", "lesson": "src/expected-masas-and-factor-types.md", "scope": "Explicit rational circle II1, rational real II-infinity, affine rational-dilation III and atomic I-infinity; all separably represented; finite matrix factors reused", "proof_locator": "Examples 4.3–4.4 and 5.1 / Corollary 5.2; preceding coefficient lesson Exercise 6.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/EM/Example-4.3", "OA-FOUND-REMAINDER/EM/Example-4.4", "OA-FOUND-REMAINDER/EM/Example-5.1", "OA-FOUND-REMAINDER/EM/Corollary-5.2" ] }, { "target_id": "TT1-V.7.Ex1(a)", "lesson": "src/regular-and-singular-maximal-abelian-algebras.md", "scope": "Atomic and nonatomic MASAs in B(H), H separable, regular; additionally mixed-case normalizer algebra fully determined", "proof_locator": "Propositions 2.1–2.2; Proposition 2.4 / Exercise 6.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/RS/Proposition-2.1", "OA-FOUND-REMAINDER/RS/Proposition-2.2", "OA-FOUND-REMAINDER/RS/Proposition-2.4", "OA-FOUND-REMAINDER/RS/Exercise-6.2" ] }, { "target_id": "TT1-V.7.Ex1(b)", "lesson": "src/regular-and-singular-maximal-abelian-algebras.md", "scope": "Free-group generator algebra diffuse MASA and singular; complete malnormal reduced-word, coefficient expectation, L2 approximation and arbitrary unitary normalizer proof", "proof_locator": "Lemmas 3.1, 4.1 and 5.1 / Proposition 4.2 / Theorem 5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/RS/Lemma-3.1", "OA-FOUND-REMAINDER/RS/Lemma-4.1", "OA-FOUND-REMAINDER/RS/Proposition-4.2", "OA-FOUND-REMAINDER/RS/Lemma-5.1", "OA-FOUND-REMAINDER/RS/Theorem-5.2" ] }, { "target_id": "TT1-V.7.6", "lesson": "src/crossed-product-coefficients-and-factor-tests.md", "scope": "Normal faithful expectation; unique coefficients and bounded matrix reconstruction; strong-star coefficient multiplication/adjoint formulas; positive sums; explicit failure of unconditional operator Fourier sums", "proof_locator": "Proposition 1.1 / Theorem 2.1 / Example 2.2 / Proposition 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CP/Proposition-1.1", "OA-FOUND-REMAINDER/CP/Theorem-2.1", "OA-FOUND-REMAINDER/CP/Example-2.2", "OA-FOUND-REMAINDER/CP/Proposition-3.1" ] }, { "target_id": "TT1-V.7.7", "lesson": "src/crossed-product-coefficients-and-factor-tests.md", "scope": "Free action iff coefficient MASA using exact stronger existing multiplier criterion", "proof_locator": "Theorem 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CP/Theorem-3.2" ] }, { "target_id": "TT1-V.7.8", "lesson": "src/crossed-product-coefficients-and-factor-tests.md", "scope": "For free actions centre equals invariant coefficient algebra, hence ergodic iff factor", "proof_locator": "Theorem 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CP/Theorem-3.2" ] }, { "target_id": "TT1-V.7.9", "lesson": "src/crossed-product-coefficients-and-factor-tests.md", "scope": "Canonical faithful normal group trace; ICC iff factor; infinite ICC factor is II1", "proof_locator": "Theorem 4.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CP/Proposition-4.1", "OA-FOUND-REMAINDER/CP/Theorem-4.2" ] }, { "target_id": "TT1-V.7.10", "lesson": "src/crossed-product-coefficients-and-factor-tests.md", "scope": "ICC definition and all four stated countable group examples with reduced-word and coordinate proofs", "proof_locator": "Section 5 / Proposition 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CP/Proposition-5.1" ] }, { "target_id": "TT1-V.7.Ex1(c)", "lesson": "src/crossed-product-coefficients-and-factor-tests.md", "scope": "Coefficient MASA regular for every free action, without requiring ergodicity", "proof_locator": "Corollary 3.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CP/Corollary-3.3" ] }, { "target_id": "TT1-IV.8.21", "lesson": "src/base-changes-and-disintegration.md", "scope": "Existence over every central subalgebra; standard probability base and factorial fibres over full centre", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Theorem-4.1" ] }, { "target_id": "TT1-IV.8.22", "lesson": "src/base-changes-and-disintegration.md", "scope": "Diagonal isomorphism gives nonsingular Borel base isomorphism; simultaneous conull inverse refinement", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Theorem-1.1" ] }, { "target_id": "TT1-IV.8.23", "lesson": "src/base-changes-and-disintegration.md", "scope": "Fibre unitary uniqueness with explicit Radon–Nikodym factor; essential bases", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Theorem-2.1" ] }, { "target_id": "TT1-IV.8.24", "lesson": "src/base-changes-and-disintegration.md", "scope": "Measurable representation field and direct integral definition", "proof_locator": "Section 3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Theorem-3.1" ] }, { "target_id": "TT1-IV.8.25", "lesson": "src/base-changes-and-disintegration.md", "scope": "Countable-algebra existence, common-null-set uniqueness and nondegeneracy", "proof_locator": "Theorem 3.1 / Corollary 3.3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Theorem-3.1", "OA-FOUND-REMAINDER/BD/Corollary-3.3" ] }, { "target_id": "TT1-IV.8.27", "lesson": "src/base-changes-and-disintegration.md", "scope": "Representation intertwiner fibre uniqueness for all algebra elements outside one common null set", "proof_locator": "Corollary 2.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Corollary-2.2" ] }, { "target_id": "TT1-IV.8.Ex2", "lesson": "src/base-changes-and-disintegration.md", "scope": "Strong convergence has one fibre-strong subsequence; stronger arbitrary sigma-finite base proof", "proof_locator": "Lemma 3.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/BD/Lemma-3.2" ] }, { "target_id": "TT1-IV.8.31", "lesson": "src/states-and-fibre-equivalence-relations.md", "scope": "Orthogonal measure iff canonical GNS map onto; measurable GNS field and full nonunital extension", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SF/Lemma-1.1", "OA-FOUND-REMAINDER/SF/Theorem-2.1" ] }, { "target_id": "TT1-IV.8.32", "lesson": "src/states-and-fibre-equivalence-relations.md", "scope": "Irreducible/MASA and factorial/centre-of-decomposable-commutant criteria", "proof_locator": "Theorem 3.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SF/Theorem-3.1" ] }, { "target_id": "TT1-IV.8.Ex3", "lesson": "src/states-and-fibre-equivalence-relations.md", "scope": "Every pair outside one common null set; actual unitary intertwiner sets agree", "proof_locator": "Theorem 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SF/Theorem-4.1" ] }, { "target_id": "TT1-IV.8.Ex4", "lesson": "src/states-and-fibre-equivalence-relations.md", "scope": "Faithful normal ambient invariance through amplification and full-carrier corner; separability of ambient Hilbert spaces proved", "proof_locator": "Theorem 5.3 and Lemmas 5.1–5.2", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SF/Lemma-5.1", "OA-FOUND-REMAINDER/SF/Lemma-5.2", "OA-FOUND-REMAINDER/SF/Theorem-5.3" ] }, { "target_id": "TT1-IV.Notes", "lesson": "src/states-and-fibre-equivalence-relations.md", "scope": null, "proof_locator": "Section7 and references", "explicit_result_ids": [ "OA-FOUND-REMAINDER/SF/Historical-notes" ] }, { "target_id": "TT1-IV.5.1", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Theorem 10.1(1–3) and own Theorem 6.1 in normal-products-and-closed-operator-graphs.md; canonical central summand and predual", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Theorem-6.1", "OA-FOUND-REMAINDER/NP-TP/Theorem-10.1" ] }, { "target_id": "TT1-IV.5.2", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Own Theorem 6.1 and Theorem 10.1; every faithful normal product realization", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Theorem-6.1", "OA-FOUND-REMAINDER/NP-TP/Theorem-10.1" ] }, { "target_id": "TT1-IV.5.3", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Corollary 8.4 and own Corollary 6.2; automatic normality declared through exact OA-MOD-NP contract", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Corollary-6.2", "OA-FOUND-REMAINDER/NP-TP/Corollary-8.4" ] }, { "target_id": "TT1-IV.5.4", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Proposition 3.1(4); two-vector amplification and positive one-vector form", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Proposition-3.1" ] }, { "target_id": "TT1-IV.5.5", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Theorem 8.2; full normal-homomorphism amplification, commutant induction and spatial isomorphism", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Theorem-8.2" ] }, { "target_id": "TT1-IV.5.6", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Corollary 8.3; common amplification and two commutant corners with central carriers one", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Corollary-8.3" ] }, { "target_id": "TT1-IV.5.7", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Lemmas 11.1–11.2; real cyclic density", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Lemma-11.1", "OA-FOUND-REMAINDER/NP-TP/Lemma-11.2" ] }, { "target_id": "TT1-IV.5.8", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Lemma 11.3; real tensor density", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Lemma-11.3" ] }, { "target_id": "TT1-IV.5.9", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Theorem 11.4; general commutation, noncyclic reduction", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Theorem-11.4" ] }, { "target_id": "TT1-IV.5.10", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Corollary 11.5(1–2), joins and intersections", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Corollary-11.5" ] }, { "target_id": "TT1-IV.5.11", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Corollary 11.5(3), centres", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Corollary-11.5" ] }, { "target_id": "TT1-IV.5.12", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Theorem 10.1(4), supports and faithfulness", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Theorem-10.1" ] }, { "target_id": "TT1-IV.5.13", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": "Normal CP tensor maps; complete preadjoint extension proof", "proof_locator": "Theorem 1.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Theorem-1.1" ] }, { "target_id": "TT1-IV.5.Ex1a", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": "Algebraic independence of commuting subfactors", "proof_locator": "Theorem 1.1 of tensor-independence-and-ideals.md", "explicit_result_ids": [ "OA-FOUND-REMAINDER/CI/Theorem-1.1" ] }, { "target_id": "TT1-IV.5.Ex1b-c", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": "Normal splitting, arbitrary nonzero product functional, arbitrary normal linear bimodule map; no positive-only reduction", "proof_locator": "Theorem 2.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Theorem-2.1" ] }, { "target_id": "TT1-IV.5.Ex3a-b", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": "Corrected affiliation criterion and four graph entries, including nondense domains", "proof_locator": "Theorems 3.1–3.2 / formula (3.2)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Closed-operator-polar-decomposition", "OA-FOUND-REMAINDER/NP/Theorem-3.1", "OA-FOUND-REMAINDER/NP/Theorem-3.2" ] }, { "target_id": "TT1-IV.5.Ex4a-c", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": "Incommensurate periodic von Neumann generation, C*-independence and no normal product functional", "proof_locator": "Proposition 4.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Proposition-4.1" ] }, { "target_id": "TT1-IV.5.Ex2", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": "Tensor MASA relative-commutant calculation", "proof_locator": "Exercise 11.7 and complete solution", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP-TP/Exercise-11.7" ] }, { "target_id": "TT1-IV.Notes.WN", "lesson": "src/normal-products-and-closed-operator-graphs.md", "scope": null, "proof_locator": "Section7", "explicit_result_ids": [ "OA-FOUND-REMAINDER/NP/Historical-tensor-notes" ] }, { "target_id": "TT1-V.7.Ex2(a)", "lesson": "src/orbit-representations-and-orthogonal-state-measures.md", "scope": "Normal implemented-commutant coefficient expectation and vector-state formula for every commutant operator; no unrestricted Fourier-sum assumption", "proof_locator": "Proposition/formulas (2.3)–(2.4)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OR/Coefficient-model-and-vector-state" ] }, { "target_id": "TT1-V.7.Ex2(b)", "lesson": "src/orbit-representations-and-orthogonal-state-measures.md", "scope": "Every support point has a norm-controlled orbit representation and pure identity-coefficient state; source pointwise domain corrected to closed invariant measure support", "proof_locator": "Theorems 3.1 and 4.1 / Exercise 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OR/Theorem-3.1", "OA-FOUND-REMAINDER/OR/Theorem-4.1", "OA-FOUND-REMAINDER/OR/Exercise-6.1" ] }, { "target_id": "TT1-V.7.Ex2(c)", "lesson": "src/orbit-representations-and-orthogonal-state-measures.md", "scope": "Complete GNS cyclicity, pure-state barycentre and multiplicative kappa map onto the original coefficient MASA", "proof_locator": "Theorem 5.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OR/Theorem-5.1" ] }, { "target_id": "TT1-V.7.Ex2(d)", "lesson": "src/orbit-representations-and-orthogonal-state-measures.md", "scope": "Same support-point orbit iff unitary equivalence, with explicit common-orbit model and disjoint-orbit zero-intertwiner proof", "proof_locator": "Theorem 4.1 / formulas (4.3)–(4.4)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/OR/Theorem-4.1" ] }, { "target_id": "TT1-IV.8.Ex1", "lesson": "imports/measurable-fields-direct-integrals.md", "scope": null, "proof_locator": "Theorem 8.2(1–3)", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-MF/Theorem-8.2" ] }, { "target_id": "TT1-V.6.1", "lesson": "imports/effros-borel-structure.md", "scope": "Exact stronger Polish projection/closed-subspace correspondence, complete specific proof read", "proof_locator": "Theorem 6.1", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-EF/Theorem-6.1" ] }, { "target_id": "TT1-V.7.1", "lesson": "providers/bf/06-kernels-and-freeness.md", "scope": "Largest identity projection and free complementary restriction", "proof_locator": "The largest identity part", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-BF06/OA-FLOW.KERNEL.PROJECTION" ] }, { "target_id": "TT1-V.7.2", "lesson": "providers/bf/06-kernels-and-freeness.md", "scope": "Freeness iff no nonzero intertwining multiplier in abelian algebra", "proof_locator": "Multiplier criterion", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-BF06/OA-FLOW.KERNEL.PROJECTION" ] }, { "target_id": "TT1-V.7.3", "lesson": "providers/bf/06-kernels-and-freeness.md", "scope": "Free and ergodic definitions; faithful ergodic abelian-group action free; effective quotient correction", "proof_locator": "The ergodic dichotomy and effective quotient", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-BF06/OA-FLOW.KERNEL.ABELIAN" ] }, { "target_id": "TT1-V.7.4", "lesson": "providers/bf/15-regular-model-independence.md", "scope": "Discrete regular crossed-product definition and covariance", "proof_locator": "Coefficient operators and left translations / R2–R3", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-BF15/OA-FLOW.REG.CONSTRUCTION", "OA-FOUND-REMAINDER/IM-BF15/OA-FLOW.REG.NORMALITY" ] }, { "target_id": "TT1-V.7.5", "lesson": "providers/bf/15-regular-model-independence.md", "scope": "Equivariant normal isomorphism of coefficients extends to normal regular-model isomorphism", "proof_locator": "The normal comparison and its generators / R13–R17", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-BF15/OA-FLOW.REG.AVERAGES", "OA-FOUND-REMAINDER/IM-BF15/OA-FLOW.REG.TENSORLOCATION", "OA-FOUND-REMAINDER/IM-BF15/OA-FLOW.REG.INDEPENDENCE", "OA-FOUND-REMAINDER/IM-BF15/OA-FLOW.REG.SYSTEMNATURALITY" ] }, { "target_id": "TT1-V.7.14", "lesson": "providers/bf/77-arbitrary-covariant-commutant.md", "scope": "Implemented commutant generation by existing E9 theorem; complete coefficient uniqueness and bounded two-coordinate strong-star reconstruction, with unordered Fourier qualification", "proof_locator": "The amplified commutant and rank-one slice / E9", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-BF77/Faithful-representations-become-equivalent-after-amplification", "OA-FOUND-REMAINDER/IM-BF77/The-amplified-commutant-and-the-rank-one-slice", "OA-FOUND-REMAINDER/OR/Coefficient-model-and-vector-state" ] }, { "target_id": "TT2-A.7", "lesson": "providers/closed-positive-forms.md", "scope": "Positive quadratic diagonal homogeneity and parallelogram identity, finite-value domain, and complete converse complex polarization on arbitrary vector spaces", "proof_locator": "OA-MOD-QF-02", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-02" ] }, { "target_id": "TT2-A.8", "lesson": "providers/closed-positive-forms.md", "scope": "Closed form/energy norm and core definition; densely defined representation with exact square-root domain and uniqueness; closability iff relative lower semicontinuity, distinguished from extension-by-infinity closedness", "proof_locator": "OA-MOD-QF-02, OA-MOD-QF-03, OA-MOD-QF-04, OA-MOD-FC-02, OA-MOD-FC-03, OA-MOD-FC-04, OA-MOD-FC-05", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-02", "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-03", "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-04", "OA-FOUND-REMAINDER/IM-FC/OA-MOD-FC-02", "OA-FOUND-REMAINDER/IM-FC/OA-MOD-FC-03", "OA-FOUND-REMAINDER/IM-FC/OA-MOD-FC-04", "OA-FOUND-REMAINDER/IM-FC/OA-MOD-FC-05" ] }, { "target_id": "TT2-A.9", "lesson": "providers/closed-positive-forms.md", "scope": "Finite form sums closed on common form domain; density retained for operator representation; form sum agrees with closure of essentially self-adjoint algebraic sum", "proof_locator": "OA-MOD-QF-10 Problem5, OA-MOD-FC-07", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-10-Problem5", "OA-FOUND-REMAINDER/IM-FC/OA-MOD-FC-07" ] }, { "target_id": "TT2-A.10", "lesson": "providers/closed-positive-forms.md", "scope": "Full form-domain/energy order iff reversed bounded resolvent order at one, hence every, positive parameter; arbitrary noncommuting self-adjoint positive operators", "proof_locator": "OA-MOD-QF-05, OA-MOD-FC-06, OA-MOD-DW-02 lemma only", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-05", "OA-FOUND-REMAINDER/IM-FC/OA-MOD-FC-06", "OA-FOUND-REMAINDER/IM-DW/OA-MOD-DW-02-lemma-only" ] }, { "target_id": "TT2-A.11", "lesson": "providers/closed-positive-forms.md", "scope": "Correct finite-energy increasing limit, density requirement, strong resolvent convergence, injective-tail logarithms and locally uniform imaginary powers", "proof_locator": "OA-MOD-QF-06, OA-MOD-QF-07, OA-MOD-QF-08, OA-MOD-QF-09", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-06", "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-07", "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-08", "OA-FOUND-REMAINDER/IM-QF/OA-MOD-QF-09" ] }, { "target_id": "TT2-A.12", "lesson": "providers/complex-interpolation.md", "scope": "Compatible pair, complete intersection and sum, bounded continuous analytic strip with endpoint decay, complete quotient interpolation norm, and exact geometric-mean endpoint theorem", "proof_locator": "OA-MOD-CI-01, OA-MOD-CI-02, OA-MOD-CI-03, OA-MOD-CI-04, OA-MOD-CI-05, OA-MOD-CI-06", "explicit_result_ids": [ "OA-FOUND-REMAINDER/IM-CI/OA-MOD-CI-01", "OA-FOUND-REMAINDER/IM-CI/OA-MOD-CI-02", "OA-FOUND-REMAINDER/IM-CI/OA-MOD-CI-03", "OA-FOUND-REMAINDER/IM-CI/OA-MOD-CI-04", "OA-FOUND-REMAINDER/IM-CI/OA-MOD-CI-05", "OA-FOUND-REMAINDER/IM-CI/OA-MOD-CI-06" ] } ] }