openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /DeterminantHomogeneity.lean
| import OAI.Analysis.IrrationalRotation.Spectral.SpectralNaturality | |
| import OAI.Analysis.IrrationalRotation.Model.ConcreteSpectral | |
| namespace OAI | |
| noncomputable section | |
| open scoped ENNReal CStarAlgebra | |
| namespace Invariant233 | |
| def positiveRegularizerScale (r : ℝ) (hr : 0 < r) : | |
| {ε : ℝ // 0 < ε} ≃ {ε : ℝ // 0 < ε} where | |
| toFun ε := ⟨r * ε.val, mul_pos hr ε.property⟩ | |
| invFun ε := ⟨ε.val / r, div_pos ε.property hr⟩ | |
| left_inv ε := by apply Subtype.ext; dsimp; field_simp [ne_of_gt hr] | |
| right_inv ε := by apply Subtype.ext; dsimp; field_simp [ne_of_gt hr] | |
| section CStar | |
| variable {A : Type*} [CStarAlgebra A] | |
| theorem star_smul_mul_smul (z : ℂ) (a : A) : | |
| star (z • a) * (z • a) = ‖z‖ ^ 2 • (star a * a) := by | |
| rw [star_smul, smul_mul_assoc, mul_smul_comm, smul_smul] | |
| rw [show star z * z = (‖z‖ ^ 2 : ℝ) by | |
| simpa only [Complex.star_def, Complex.ofReal_pow] using Complex.conj_mul' z] | |
| exact (RCLike.real_smul_eq_coe_smul (K := ℂ) _ _).symm | |
| theorem regularizedLog_smul (z : ℂ) (hz : z ≠ 0) (a : A) (ε : ℝ) (hε : 0 < ε) : | |
| CFC.log (star (z • a) * (z • a) + (‖z‖ ^ 2 * ε) • (1 : A)) = | |
| algebraMap ℝ A (2 * Real.log ‖z‖) + | |
| CFC.log (star a * a + ε • (1 : A)) := by | |
| rw [star_smul_mul_smul, mul_smul, ← smul_add] | |
| rw [CFC.log_smul _ (fun x hx => ne_of_gt (regularizedNormSquare_spectrum_pos a ε hε x hx)) | |
| (pow_ne_zero _ (norm_ne_zero_iff.mpr hz)) (by simp [IsSelfAdjoint])] | |
| rw [Real.log_pow] | |
| norm_num | |
| theorem fugledeKadisonDeterminant_smul (τ : TracialState A) (z : ℂ) (a : A) : | |
| fugledeKadisonDeterminant τ (z • a) = | |
| ENNReal.ofReal ‖z‖ * fugledeKadisonDeterminant τ a := by | |
| by_cases hz : z = 0 | |
| · subst z | |
| simp [fugledeKadisonDeterminant_zero] | |
| have hn : 0 < ‖z‖ := norm_pos_iff.mpr hz | |
| unfold fugledeKadisonDeterminant | |
| rw [← (positiveRegularizerScale (‖z‖ ^ 2) (sq_pos_of_pos hn)).iInf_comp] | |
| rw [ENNReal.mul_iInf_of_ne (ENNReal.ofReal_ne_zero_iff.mpr hn) ENNReal.ofReal_ne_top] | |
| apply iInf_congr | |
| intro ε | |
| change ENNReal.ofReal (Real.exp ((τ (CFC.log | |
| (star (z • a) * (z • a) + (‖z‖ ^ 2 * ε.val) • (1 : A)))).re / 2)) = _ | |
| rw [regularizedLog_smul z hz a ε.val ε.property] | |
| rw [map_add, TracialState.map_real] | |
| simp only [Complex.add_re, Complex.ofReal_re] | |
| rw [show (2 * Real.log ‖z‖ + (τ (CFC.log (star a * a + ε.val • (1 : A)))).re) / 2 = | |
| Real.log ‖z‖ + (τ (CFC.log (star a * a + ε.val • (1 : A)))).re / 2 by ring] | |
| rw [Real.exp_add, Real.exp_log hn, ENNReal.ofReal_mul hn.le] | |
| end CStar | |
| section Vector | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| omit [CompleteSpace H] in | |
| theorem vectorTrace_real_scalar (ξ : H) (hξ : ‖ξ‖ = 1) (r : ℝ) : | |
| vectorTrace ξ (algebraMap ℝ (H →L[ℂ] H) r) = (r : ℂ) := by | |
| simp [vectorTrace, Algebra.algebraMap_eq_smul_one, inner_smul_right_eq_smul, | |
| inner_self_eq_norm_sq_to_K, hξ, Complex.real_smul] | |
| theorem vectorFugledeKadisonDeterminant_zero (ξ : H) (hξ : ‖ξ‖ = 1) : | |
| vectorFugledeKadisonDeterminant ξ 0 = 0 := by | |
| unfold vectorFugledeKadisonDeterminant | |
| simp only [star_zero, zero_mul, zero_add, ← Algebra.algebraMap_eq_smul_one, | |
| CFC.log_algebraMap, vectorTrace_real_scalar ξ hξ, Complex.ofReal_re] | |
| have h : ∀ ε : {r : ℝ // 0 < r}, | |
| Real.exp (Real.log ε.val / 2) = Real.sqrt ε.val := | |
| fun ε => exp_half_log_eq_sqrt ε.property | |
| simp_rw [h] | |
| simpa using iInf_sqrt_sq_add_pos 0 le_rfl | |
| theorem vectorFugledeKadisonDeterminant_smul (ξ : H) (hξ : ‖ξ‖ = 1) | |
| (z : ℂ) (T : H →L[ℂ] H) : | |
| vectorFugledeKadisonDeterminant ξ (z • T) = | |
| ENNReal.ofReal ‖z‖ * vectorFugledeKadisonDeterminant ξ T := by | |
| by_cases hz : z = 0 | |
| · subst z | |
| simp [vectorFugledeKadisonDeterminant_zero ξ hξ] | |
| have hn : 0 < ‖z‖ := norm_pos_iff.mpr hz | |
| unfold vectorFugledeKadisonDeterminant | |
| rw [← (positiveRegularizerScale (‖z‖ ^ 2) (sq_pos_of_pos hn)).iInf_comp] | |
| rw [ENNReal.mul_iInf_of_ne (ENNReal.ofReal_ne_zero_iff.mpr hn) ENNReal.ofReal_ne_top] | |
| apply iInf_congr | |
| intro ε | |
| change ENNReal.ofReal (Real.exp ((vectorTrace ξ (CFC.log | |
| (star (z • T) * (z • T) + (‖z‖ ^ 2 * ε.val) • (1 : H →L[ℂ] H)))).re / 2)) = _ | |
| rw [regularizedLog_smul z hz T ε.val ε.property] | |
| have hadd (A B : H →L[ℂ] H) : vectorTrace ξ (A + B) = vectorTrace ξ A + vectorTrace ξ B := by | |
| simp [vectorTrace, inner_add_right] | |
| rw [hadd, vectorTrace_real_scalar ξ hξ] | |
| simp only [Complex.add_re, Complex.ofReal_re] | |
| rw [show (2 * Real.log ‖z‖ + (vectorTrace ξ (CFC.log (star T * T + ε.val • (1 : H →L[ℂ] H)))).re) / 2 = | |
| Real.log ‖z‖ + (vectorTrace ξ (CFC.log (star T * T + ε.val • (1 : H →L[ℂ] H)))).re / 2 by ring] | |
| rw [Real.exp_add, Real.exp_log hn, ENNReal.ofReal_mul hn.le] | |
| end Vector | |
| end Invariant233 | |
| end | |
| end OAI | |