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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | 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
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