openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /DeterminantWeightBridge.lean
| import OAI.Analysis.IrrationalRotation.Spectral.ScalarDeterminantZero | |
| import OAI.Analysis.IrrationalRotation.Rotation.AnalyticWeightBridge | |
| import OAI.Analysis.IrrationalRotation.Spectral.SpectralNaturality | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace Invariant233 | |
| theorem ScalarDeterminant.regularized_determinant_eq_zero_of_log_integral | |
| {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ] | |
| {f : X → ℝ} (hm : Measurable f) (hbound : ∀ x, 0 ≤ f x ∧ f x ≤ 1) | |
| (hpos : ∀ᵐ x ∂μ, 0 < f x) (hinf : HasNegInfiniteLogIntegral μ f) : | |
| (⨅ ε : {r : ℝ // 0 < r}, ENNReal.ofReal | |
| (Real.exp ((∫ x, Real.log (f x ^ 2 + ε.val) ∂μ) / 2))) = 0 := by | |
| apply ScalarDeterminant.regularized_determinant_eq_zero hm hbound | |
| intro B | |
| obtain ⟨M, hM, hlt⟩ := exists_pos_truncated_log_integral_lt f hm | |
| (fun x => (hbound x).1) (fun x => (hbound x).2) hpos | |
| ((hasNegInfiniteLogIntegral_iff_of_ae_pos f hpos).mp hinf) (-B / 2) | |
| refine ⟨M, hM, ?_⟩ | |
| linarith | |
| theorem IsContinuousCircleWeight.regularized_determinant_eq_zero | |
| {f : Circle → ℝ} (hf : IsContinuousCircleWeight f) : | |
| (⨅ ε : {r : ℝ // 0 < r}, ENNReal.ofReal | |
| (Real.exp ((∫ x, Real.log (f x ^ 2 + ε.val) ∂circleMeasure) / 2))) = 0 := by | |
| let : IsProbabilityMeasure circleMeasure := ⟨by | |
| simp only [circleMeasure, AddCircle.measure_univ, ENNReal.ofReal_one]⟩ | |
| exact ScalarDeterminant.regularized_determinant_eq_zero_of_log_integral | |
| hf.1.measurable hf.2.1 hf.ae_pos hf.2.2.2 | |
| variable {A : Type*} [CStarAlgebra A] | |
| theorem fugledeKadisonDeterminant_continuous_circle_weight | |
| (π : C(Circle, ℂ) →⋆ₐ[ℂ] A) (τ : TracialState A) | |
| (hτ : ∀ g : C(Circle, ℂ), τ (π g) = ∫ x, g x ∂circleMeasure) | |
| (f : C(Circle, ℂ)) (hf : IsContinuousCircleWeight (fun x => ‖f x‖)) : | |
| fugledeKadisonDeterminant τ (π f) = 0 := by | |
| rw [fugledeKadisonDeterminant_continuous_representation circleMeasure π τ hτ] | |
| exact hf.regularized_determinant_eq_zero | |
| theorem fugledeKadisonDeterminant_real_circle_weight | |
| (π : C(Circle, ℂ) →⋆ₐ[ℂ] A) (τ : TracialState A) | |
| (hτ : ∀ g : C(Circle, ℂ), τ (π g) = ∫ x, g x ∂circleMeasure) | |
| (f : C(Circle, ℝ)) (hf : IsContinuousCircleWeight f) : | |
| fugledeKadisonDeterminant τ | |
| (π ⟨fun x => (f x : ℂ), Complex.continuous_ofReal.comp f.continuous⟩) = 0 := by | |
| apply fugledeKadisonDeterminant_continuous_circle_weight π τ hτ | |
| convert hf using 1 | |
| funext x | |
| simp only [ContinuousMap.coe_mk, Complex.norm_real, Real.norm_eq_abs, | |
| abs_of_nonneg (hf.2.1 x).1] | |
| theorem fugledeKadisonDeterminant_unitary_real_circle_weight | |
| (π : C(Circle, ℂ) →⋆ₐ[ℂ] A) (τ : TracialState A) | |
| (hτ : ∀ g : C(Circle, ℂ), τ (π g) = ∫ x, g x ∂circleMeasure) | |
| (u : unitary A) (f : C(Circle, ℝ)) (hf : IsContinuousCircleWeight f) : | |
| fugledeKadisonDeterminant τ | |
| ((u : A) * π ⟨fun x => (f x : ℂ), Complex.continuous_ofReal.comp f.continuous⟩) = 0 := by | |
| rw [fugledeKadisonDeterminant_unitary_mul] | |
| exact fugledeKadisonDeterminant_real_circle_weight π τ hτ f hf | |
| end Invariant233 | |
| end | |
| end OAI | |