openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /CircleVectorDeterminant.lean
| import OAI.Analysis.IrrationalRotation.Spectral.VectorSpectral | |
| import OAI.Analysis.IrrationalRotation.Spectral.DeterminantWeightBridge | |
| import OAI.Analysis.IrrationalRotation.Rotation.Rotation | |
| import OAI.Analysis.IrrationalRotation.Rotation.RotationDiagonalState | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| namespace Invariant233.Rotation.SpectralCompatibility | |
| theorem vectorDeterminant_representation (θ : ℝ) (a : C(Circle, ℂ)) : | |
| vectorFugledeKadisonDeterminant vacuum (representation θ a) = | |
| ⨅ ε : {r : ℝ // 0 < r}, ENNReal.ofReal | |
| (Real.exp ((∫ x, Real.log (‖a x‖ ^ 2 + ε.val) ∂circleMeasure) / 2)) := by | |
| exact vectorFugledeKadisonDeterminant_of_continuous_integration circleMeasure | |
| (representationStarAlgHom θ) vacuum (vacuumFunctional_representation θ) a | |
| theorem vectorDeterminant_weightedReal_eq_zero (θ : ℝ) (f : C(Circle, ℝ)) | |
| (hf : IsContinuousCircleWeight f) : | |
| vectorFugledeKadisonDeterminant vacuum (weightedReal θ f) = 0 := by | |
| unfold weightedReal weighted | |
| rw [vectorFugledeKadisonDeterminant_isometry_mul vacuum U | |
| (representation θ (complexWeight f)) (Unitary.star_mul_self_of_mem U_mem_unitary)] | |
| rw [vectorDeterminant_representation] | |
| have hn : ∀ x, ‖complexWeight f x‖ = f x := by | |
| intro x | |
| simp only [complexWeight, ContinuousMap.coe_mk, Complex.norm_real, | |
| Real.norm_eq_abs, abs_of_nonneg (hf.2.1 x).1] | |
| simp_rw [hn] | |
| exact hf.regularized_determinant_eq_zero | |
| end Invariant233.Rotation.SpectralCompatibility | |
| end | |
| end OAI | |