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9655094 | 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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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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