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9eeb9c7 | 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 | import OAI.Analysis.IrrationalRotation.Rotation.ContinuousWeightOperatorBridge
import OAI.Analysis.IrrationalRotation.Rotation.RotationProjectionTransport
namespace OAI
noncomputable section
open MeasureTheory
namespace Invariant233.CircleParameterLaws
variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p)
include hp
theorem continuousWeight_hasOnlyTrivialInvariantProjections
(d : BackgroundBumpData θ)
(hstar : HasOnlyTrivialInvariantProjections (Rotation.algebra θ)
(Rotation.U * Rotation.measurableRepresentation θ
(fun x => (p.measurableWeight θ x : ℂ)) 1 zero_le_one
hp.measurable_complexParameterWeight hp.norm_complexParameterWeight_le_one)) :
HasOnlyTrivialInvariantProjections (Rotation.algebra θ)
(Rotation.weightedReal θ
⟨p.continuousWeight θ d, (hp.isContinuousCircleWeight_continuousWeight d).1⟩) := by
let f : C(Circle, ℝ) :=
⟨p.continuousWeight θ d, (hp.isContinuousCircleWeight_continuousWeight d).1⟩
let H := Rotation.exponentialTransfer θ (p.continuousTransfer θ d)
let Tstar := Rotation.U * Rotation.measurableRepresentation θ
(fun x => (p.measurableWeight θ x : ℂ)) 1 zero_le_one
hp.measurable_complexParameterWeight hp.norm_complexParameterWeight_le_one
have h := hp.continuousWeight_natural_intertwining d
let hD : ∀ ξ : H.domain, Rotation.weightedReal θ f ξ.val ∈ H.domain :=
fun ξ => (h ξ).choose
have hI : ∀ ξ : H.domain, H ⟨Rotation.weightedReal θ f ξ.val, hD ξ⟩ =
(Real.exp (-1) : ℂ) • Tstar (H ξ) := fun ξ => (h ξ).choose_spec
exact Rotation.hasOnlyTrivialInvariantProjections_of_scaled_exponentialTransfer θ
(measurable_continuousTransfer d) (Rotation.weightedReal θ f) Tstar
(Real.exp (-1) : ℂ) (Complex.ofReal_ne_zero.mpr (Real.exp_ne_zero _)) hD hI hstar
end Invariant233.CircleParameterLaws
end
end OAI
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