| import Mathlib |
| import OAI.Analysis.BrownMeasure.Polar |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology ENNReal CStarAlgebra |
| namespace FiniteFactor |
| local instance : Module ℝ ProductOp := Module.complexToReal ProductOp |
| local instance : IsometricContinuousFunctionalCalculus ℝ ProductOp IsSelfAdjoint := |
| IsSelfAdjoint.instIsometricContinuousFunctionalCalculus |
|
|
| lemma sourceShift_eq_transport (θ : SourceCoordinate → ℝ) : |
| sourceShift SourceCoordinate=sourceTransport θ 0 1 := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourceShift_apply_ae u,sourceTransport_apply_ae θ 0 1 u,sourceHilbert_cube] with z hz ht hc |
| rw [hz,ht] |
| congr 1 |
| apply Prod.ext |
| · funext i |
| change z.1 i=Int.fract (z.1 i+(0:ℤ)*θ i) |
| simpa using (Int.fract_eq_self.mpr (hc i)).symm |
| · rfl |
|
|
| lemma sourceShift_unitary : star (sourceShift SourceCoordinate)*sourceShift SourceCoordinate=1 ∧ |
| sourceShift SourceCoordinate*star (sourceShift SourceCoordinate)=1 := by |
| rw [sourceShift_eq_transport (fun _=>0),sourceTransport_star,sourceTransport_mul,sourceTransport_mul] |
| simp [sourceTransport_zero] |
|
|
| def sourceWeightSqrtBase (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : SourceBaseFunction where |
| toFun := fun x=>(Real.sqrt (sourceCocycle θ sourceLogWeight 1 x):ℂ) |
| measurable := Complex.measurable_ofReal.comp (Real.continuous_sqrt.measurable.comp |
| (sourceCocycle_measurable θ sourceLogWeight 1)) |
| bound := 1 |
| bound_nonneg := zero_le_one |
| norm_le := by |
| intro x hx |
| rw [Complex.norm_real,Real.norm_of_nonneg (Real.sqrt_nonneg _)] |
| exact (Real.sqrt_le_one).mpr (sourceWeight_le_one θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg hx) |
|
|
| lemma sourceWeight_positive (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| 0 ≤ sourcePi θ (sourceWeightBase θ hθ) := by |
| let B := sourcePi θ (sourceWeightSqrtBase θ hθ) |
| have he : star B*B=sourcePi θ (sourceWeightBase θ hθ) := by |
| dsimp [B] |
| rw [sourcePi_star] |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourcePi_apply_ae θ (sourceBaseStar (sourceWeightSqrtBase θ hθ)) |
| (sourcePi θ (sourceWeightSqrtBase θ hθ) u),sourcePi_apply_ae θ (sourceWeightSqrtBase θ hθ) u, |
| sourcePi_apply_ae θ (sourceWeightBase θ hθ) u] with z h1 h2 h3 |
| change (sourcePi θ (sourceBaseStar (sourceWeightSqrtBase θ hθ)) |
| (sourcePi θ (sourceWeightSqrtBase θ hθ) u)) z=(sourcePi θ (sourceWeightBase θ hθ) u) z |
| rw [h1,h2,h3] |
| simp only [sourcePiSymbol,sourceBaseStar,sourceWeightSqrtBase,sourceWeightBase, |
| Complex.star_def,Complex.conj_ofReal,← mul_assoc,← Complex.ofReal_mul, |
| Real.mul_self_sqrt (sourceCocycle_pos _ _ _ _).le] |
| rw [← he] |
| exact star_mul_self_nonneg B |
|
|
| lemma sourceOperator_abs (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| CFC.abs (sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg)= |
| sourcePi θ (sourceWeightBase θ hθ) := by |
| let P := sourcePi θ (sourceWeightBase θ hθ) |
| have hp : 0 ≤ P := sourceWeight_positive θ hθ |
| change CFC.sqrt (star (sourceShift SourceCoordinate*P)*(sourceShift SourceCoordinate*P))=P |
| have he : star (sourceShift SourceCoordinate*P)*(sourceShift SourceCoordinate*P)=P*P := by |
| rw [star_mul] |
| calc |
| star P * star (sourceShift SourceCoordinate) * (sourceShift SourceCoordinate * P) = |
| star P * ((star (sourceShift SourceCoordinate)*sourceShift SourceCoordinate)*P) := by |
| noncomm_ring |
| _ = P*P := by rw [sourceShift_unitary.1,one_mul,hp.star_eq] |
| rw [he,CFC.sqrt_mul_self P hp] |
|
|
| lemma sourcePolarFamily_zero (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) : |
| sourcePolarFamily θ hθ 0=sourcePi θ (sourceWeightBase θ hθ) := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| filter_upwards [sourcePi_apply_ae θ (sourcePolarBase θ hθ 0) u, |
| sourcePi_apply_ae θ (sourceWeightBase θ hθ) u] with z h1 h2 |
| change (sourcePi θ (sourcePolarBase θ hθ 0) u) z=_ |
| rw [h1,h2] |
| simp [sourcePiSymbol,sourcePolarBase,sourceWeightBase] |
| left |
| rw [coordinateRotate_add,add_zero] |
|
|
| lemma sourceTransport_conjugate_pi (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ) (n : ℤ) : |
| sourceTransport θ 0 n * sourcePi θ (sourceWeightBase θ hθ)*sourceTransport θ 0 (-n)= |
| sourcePolarFamily θ hθ n := by |
| apply ContinuousLinearMap.ext |
| intro u |
| apply Lp.ext |
| have ht := (sourceTransportIndex_preserving θ 0 n).quasiMeasurePreserving.ae |
| (sourcePi_apply_ae θ (sourceWeightBase θ hθ) (sourceTransport θ 0 (-n) u)) |
| have hu := (sourceTransportIndex_preserving θ 0 n).quasiMeasurePreserving.ae |
| (sourceTransport_apply_ae θ 0 (-n) u) |
| filter_upwards [sourceTransport_apply_ae θ 0 n |
| (sourcePi θ (sourceWeightBase θ hθ) (sourceTransport θ 0 (-n) u)),ht,hu, |
| sourcePi_apply_ae θ (sourcePolarBase θ hθ n) u,sourceHilbert_cube] with z h1 h2 h3 h4 hc |
| change (sourceTransport θ 0 n (sourcePi θ (sourceWeightBase θ hθ) (sourceTransport θ 0 (-n) u))) z=_ |
| change (sourceTransport θ 0 n (sourcePi θ (sourceWeightBase θ hθ) (sourceTransport θ 0 (-n) u))) z= |
| (sourcePi θ (sourcePolarBase θ hθ n) u) z |
| rw [h1,h2,h3,h4] |
| congr 1 |
| · dsimp [sourcePiSymbol,sourceWeightBase,sourcePolarBase,sourceTransportIndex] |
| congr 2 |
| rw [coordinateRotate_add,coordinateRotate_add] |
| congr 1 |
| ring |
| · congr 1 |
| apply Prod.ext |
| · funext i |
| simpa [sourceTransportIndex,coordinateRotate] using Int.fract_eq_self.mpr (hc i) |
| · simp [sourceTransportIndex] |
| end FiniteFactor |
| end |
|
|
| end OAI |
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