openAI-math-dataset / lean /OAI /Analysis /JiangSu /DimensionFunctions /FiniteSpectralIntegralCommuting.lean
| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegralAbelian | |
| import OAI.Analysis.JiangSu.MatrixAlgebras.PositiveContinuousRepresentation | |
| namespace OAI | |
| noncomputable section | |
| namespace Pureness | |
| open TraceCone Set | |
| open scoped NNReal ENNReal ComplexOrder IsMulCommutative | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| theorem finiteSpectralIntegral_add_commute | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M))) | |
| (hfinite : ∀ x, d x ≠ ⊤) | |
| (n : ℕ) (M N : Positive (FinMatrix A n)) (hMN : Commute M.1 N.1) : | |
| finiteSpectralIntegral d n (Positive.add M N) = | |
| finiteSpectralIntegral d n M + finiteSpectralIntegral d n N := by | |
| let S : StarSubalgebra ℂ (FinMatrix A n) := (StarAlgebra.adjoin ℂ {M.1,N.1}).topologicalClosure | |
| let : IsClosed (S : Set (FinMatrix A n)) := (StarAlgebra.adjoin ℂ {M.1,N.1}).isClosed_topologicalClosure | |
| have hcomm : ∀ x ∈ ({M.1,N.1} : Set (FinMatrix A n)), ∀ y ∈ ({M.1,N.1} : Set (FinMatrix A n)), x*y = y*x := by | |
| intro x hx y hy | |
| simp only [mem_insert_iff,mem_singleton_iff] at hx hy | |
| rcases hx with rfl | rfl <;> rcases hy with rfl | rfl | |
| · rfl | |
| · exact hMN.eq | |
| · exact hMN.eq.symm | |
| · rfl | |
| let : IsMulCommutative (StarAlgebra.adjoin ℂ {M.1,N.1}) := | |
| StarAlgebra.isMulCommutative_adjoin ℂ (by | |
| intro element helement | |
| simp only [mem_insert_iff,mem_singleton_iff] at helement | |
| rcases helement with rfl | rfl | |
| · exact M.2.isSelfAdjoint.isStarNormal | |
| · exact N.2.isSelfAdjoint.isStarNormal) | |
| (fun left hleft right hright _ => hcomm left hleft right hright) (by | |
| intro x hx y hy _ | |
| have hyStar : star y = y := by | |
| simp only [mem_insert_iff,mem_singleton_iff] at hy | |
| rcases hy with rfl | rfl | |
| · exact M.2.star_eq | |
| · exact N.2.star_eq | |
| rw [hyStar] | |
| exact hcomm x hx y hy) | |
| let : IsMulCommutative S := (StarAlgebra.adjoin ℂ {M.1,N.1}).isMulCommutative_topologicalClosure | |
| let : CommCStarAlgebra S := { } | |
| have hMS : M.1 ∈ S := (StarAlgebra.adjoin ℂ {M.1,N.1}).le_topologicalClosure | |
| (StarAlgebra.subset_adjoin ℂ _ (by simp)) | |
| have hNS : N.1 ∈ S := (StarAlgebra.adjoin ℂ {M.1,N.1}).le_topologicalClosure | |
| (StarAlgebra.subset_adjoin ℂ _ (by simp)) | |
| let X := WeakDual.characterSpace ℂ S | |
| let e := gelfandStarTransform S | |
| let φ : C(X, ℂ) →⋆ₙₐ[ℂ] FinMatrix A n := | |
| (S.subtype.comp e.symm.toStarAlgHom).toNonUnitalStarAlgHom | |
| have hφ : Function.Injective φ := Subtype.val_injective.comp e.symm.injective | |
| let m : C(X, ℂ) := e ⟨M.1,hMS⟩ | |
| let q : C(X, ℂ) := e ⟨N.1,hNS⟩ | |
| have hm : φ m = M.1 := by change (e.symm (e ⟨M.1,hMS⟩)).1 = M.1; rw [e.symm_apply_apply] | |
| have hq : φ q = N.1 := by change (e.symm (e ⟨N.1,hNS⟩)).1 = N.1; rw [e.symm_apply_apply] | |
| have hmpos : 0 ≤ m := (nonUnitalStarAlgHom_nonneg_iff φ hφ m).mp (hm.symm ▸ M.2) | |
| have hqpos : 0 ≤ q := (nonUnitalStarAlgHom_nonneg_iff φ hφ q).mp (hq.symm ▸ N.2) | |
| obtain ⟨f,hf⟩ := continuousNNRealPositive_surjective (⟨m,hmpos⟩ : Positive C(X, ℂ)) | |
| obtain ⟨g,hg⟩ := continuousNNRealPositive_surjective (⟨q,hqpos⟩ : Positive C(X, ℂ)) | |
| have hfM : Positive.map φ (continuousNNRealPositive f) = M := by rw [hf]; exact Subtype.ext hm | |
| have hgN : Positive.map φ (continuousNNRealPositive g) = N := by rw [hg]; exact Subtype.ext hq | |
| simpa only [hfM,hgN] using finiteSpectralIntegral_image_add d hd hlsc hfinite n φ f g | |
| end Pureness | |
| namespace Pureness | |
| open TraceCone MeasureTheory Set | |
| open scoped ENNReal NNReal | |
| variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D] | |
| theorem positive_cut_smul (r : ℝ≥0) (hr : 0 < r) (ε : ℝ≥0) (M : Positive D) : | |
| Positive.cut ε (Positive.smul r M) = Positive.smul r (Positive.cut (ε/r) M) := by | |
| apply Subtype.ext | |
| change cfcₙ (fun t : ℝ => max (t-ε) 0) ((r : ℝ) • M.1) = | |
| (r : ℝ) • cfcₙ (fun t : ℝ => max (t-(ε/r : ℝ≥0)) 0) M.1 | |
| rw [← cfcₙ_comp_smul (r : ℝ) (fun t : ℝ => max (t-ε) 0) M.1, | |
| ← cfcₙ_smul (r : ℝ) (fun t : ℝ => max (t-(ε/r : ℝ≥0)) 0) M.1 (h0 := by simp only [zero_sub]; exact max_eq_right (neg_nonpos.mpr (NNReal.coe_nonneg _)))] | |
| apply cfcₙ_congr | |
| intro t ht | |
| change max ((r : ℝ)*t-ε) 0 = (r : ℝ)*max (t-(ε/r : ℝ≥0)) 0 | |
| rw [NNReal.coe_div, mul_max_of_nonneg _ _ (show (0 : ℝ) ≤ r from r.2),mul_zero,mul_sub] | |
| congr 1 | |
| field_simp | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| lemma finitePositiveClass_smul (n : ℕ) (M : Positive (FinMatrix A n)) | |
| (r : ℝ≥0) (hr : 0 < r) : | |
| finitePositiveClass n (Positive.smul r M) = finitePositiveClass n M := by | |
| have h := positive_smul_cuntz_equiv M.1 r hr | |
| exact le_antisymm ((finitePositiveClass_le_iff n _ _).mpr h.1) | |
| ((finitePositiveClass_le_iff n _ _).mpr h.2) | |
| lemma finiteDimensionDistribution_smul | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (r : ℝ≥0) (hr : 0 < r) (t : ℝ) : | |
| finiteDimensionDistribution d n (Positive.smul r M) t = | |
| finiteDimensionDistribution d n M (t/r) := by | |
| unfold finiteDimensionDistribution | |
| rw [positive_cut_smul r hr,finitePositiveClass_smul n _ r hr] | |
| congr 3 | |
| exact by | |
| by_cases ht : 0 ≤ t | |
| · simp [Real.toNNReal_div ht, Real.toNNReal_coe] | |
| · have htn : t ≤ 0 := (lt_of_not_ge ht).le | |
| rw [Real.toNNReal_eq_zero.mpr htn, zero_div] | |
| exact (Real.toNNReal_eq_zero.mpr (div_nonpos_of_nonpos_of_nonneg htn r.2)).symm | |
| lemma positive_halfline_integral_div (f : ℝ → ℝ≥0∞) (hf : Measurable f) | |
| (r : ℝ) (hr : 0 < r) : | |
| (∫⁻ t in Ioi (0 : ℝ), f (t/r)) = ENNReal.ofReal r * ∫⁻ t in Ioi (0 : ℝ), f t := by | |
| let g := (Ioi (0 : ℝ)).indicator f | |
| have hg : Measurable g := hf.indicator measurableSet_Ioi | |
| have heq : (Ioi (0 : ℝ)).indicator (fun t => f (t/r)) = fun t => g (r⁻¹*t) := by | |
| funext t | |
| by_cases ht : 0 < t | |
| · simp [g,ht,inv_mul_eq_div,div_pos ht hr] | |
| · have hn : ¬ 0 < t/r := by simpa only [div_pos_iff_of_pos_right hr] using ht | |
| simp [g,ht,inv_mul_eq_div,hn] | |
| rw [← lintegral_indicator measurableSet_Ioi, heq, ← lintegral_map hg (show Measurable (fun t : ℝ => r⁻¹*t) from measurable_const.mul measurable_id), | |
| Real.map_volume_mul_left (inv_ne_zero hr.ne'), lintegral_smul_measure] | |
| simp only [inv_inv, abs_of_pos hr, smul_eq_mul] | |
| rw [lintegral_indicator measurableSet_Ioi] | |
| theorem finiteSpectralIntegral_smul | |
| (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d) | |
| (n : ℕ) (M : Positive (FinMatrix A n)) (r : ℝ≥0) : | |
| finiteSpectralIntegral d n (Positive.smul r M) = (r : ℝ≥0∞)*finiteSpectralIntegral d n M := by | |
| by_cases hr : r = 0 | |
| · subst r | |
| have hzero : Positive.smul 0 M = Positive.zero := by apply Subtype.ext; change (0 : ℝ) • M.1 = 0; exact zero_smul ℝ M.1 | |
| simp [hzero,finiteSpectralIntegral_zero] | |
| · have hrpos : 0 < r := pos_iff_ne_zero.mpr hr | |
| simp only [finiteSpectralIntegral,finiteDimensionDistribution_smul d n M r hrpos] | |
| rw [positive_halfline_integral_div _ (finiteDimensionDistribution_measurable d hd n M) r hrpos, | |
| ENNReal.ofReal_coe_nnreal] | |
| end Pureness | |
| end | |
| end OAI | |