| import OAI.Analysis.IrrationalRotation.Product.ProductMultiplierFunctionalCalculus |
| import OAI.Analysis.IrrationalRotation.Product.ProductScalarity |
| import OAI.Analysis.IrrationalRotation.Product.ProductTracialVector |
| import OAI.Analysis.IrrationalRotation.Analysis.DiffuseFromSmallPartitions |
| import Mathlib.MeasureTheory.Measure.Regular |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology ENNReal |
| namespace Invariant233.ProductModel |
|
|
| def borelSetProjection (θ : ℕ → ℝ) (s : Set Base) (hs : MeasurableSet s) : |
| Hilbert →L[ℂ] Hilbert := |
| multiplier θ (s.indicator (fun _ => (1 : ℂ))) 1 zero_le_one |
| (measurable_const.indicator hs) (by |
| intro x |
| by_cases hx : x ∈ s <;> simp [hx]) |
|
|
| theorem borelSetProjection_mem (θ : ℕ → ℝ) (s : Set Base) (hs : MeasurableSet s) : |
| borelSetProjection θ s hs ∈ factor θ := |
| subset_vonNeumannGenerated _ (Or.inr ⟨_, _, _, _, _, rfl⟩) |
|
|
| theorem borelSetProjection_isStarProjection (θ : ℕ → ℝ) (s : Set Base) |
| (hs : MeasurableSet s) : IsStarProjection (borelSetProjection θ s hs) := by |
| constructor |
| · change borelSetProjection θ s hs * borelSetProjection θ s hs = borelSetProjection θ s hs |
| symm |
| apply WeightedShift.fieldDiagonal_mul_of_coeff_eq |
| exact Eventually.of_forall fun x k => by |
| change s.indicator (fun _ => (1 : ℂ)) (rotationIterate θ k x) = |
| s.indicator (fun _ => (1 : ℂ)) (rotationIterate θ k x) * |
| s.indicator (fun _ => (1 : ℂ)) (rotationIterate θ k x) |
| by_cases hx : rotationIterate θ k x ∈ s <;> simp [hx] |
| · change star (borelSetProjection θ s hs) = borelSetProjection θ s hs |
| symm |
| apply WeightedShift.fieldDiagonal_star_of_eq |
| exact Eventually.of_forall fun x => lp.ext (funext fun k => by |
| change s.indicator (fun _ => (1 : ℂ)) (rotationIterate θ k x) = |
| star (s.indicator (fun _ => (1 : ℂ)) (rotationIterate θ k x)) |
| by_cases hx : rotationIterate θ k x ∈ s <;> simp [hx]) |
|
|
| theorem vectorTrace_borelSetProjection (θ : ℕ → ℝ) (s : Set Base) (hs : MeasurableSet s) : |
| vectorTrace traceVector (borelSetProjection θ s hs) = (baseMeasure.real s : ℂ) := by |
| unfold borelSetProjection |
| rw [vectorTrace_multiplier, integral_indicator hs, integral_const] |
| simp [Measure.real] |
|
|
| theorem borelSetProjection_joint_kernel {ι : Type*} [Fintype ι] |
| (θ : ℕ → ℝ) (s : ι → Set Base) (hs : ∀ i, MeasurableSet (s i)) |
| (hcover : ∀ x : Base, ∃ i, x ∈ s i) (v : Hilbert) |
| (hv : ∀ i, borelSetProjection θ (s i) (hs i) v = 0) : v = 0 := by |
| have hcoord : ∀ i, ∀ᵐ x ∂baseMeasure, ∀ k : ℤ, |
| (s i).indicator (fun _ => (1 : ℂ)) (rotationIterate θ k x) * v x k = 0 := by |
| intro i |
| have h := multiplier_coordinates_ae θ |
| ((s i).indicator (fun _ => (1 : ℂ))) 1 zero_le_one |
| (measurable_const.indicator (hs i)) (by intro x; by_cases hx : x ∈ s i <;> simp [hx]) v |
| change ∀ᵐ x ∂baseMeasure, ∀ k, |
| borelSetProjection θ (s i) (hs i) v x k = _ at h |
| rw [hv i] at h |
| filter_upwards [h, Lp.coeFn_zero Fiber 2 baseMeasure] with x hx hz |
| intro k |
| simpa [hz] using (hx k).symm |
| apply Lp.ext |
| filter_upwards [ae_all_iff.mpr hcoord, Lp.coeFn_zero Fiber 2 baseMeasure] with x hx hz |
| rw [hz] |
| apply lp.ext |
| funext k |
| obtain ⟨i, hi⟩ := hcover (rotationIterate θ k x) |
| simpa [hi] using hx i k |
|
|
| theorem factor_isDiffuse (θ : ℕ → ℝ) : IsDiffuseVonNeumannAlgebra (factor θ) := by |
| classical |
| apply isDiffuse_of_small_trace_separating_families (factor θ) traceVector (traceVector_faithfulTracial θ) |
| intro ε hε |
| have hsmall (x : Base) : ∃ U : Set Base, x ∈ U ∧ IsOpen U ∧ baseMeasure U < ENNReal.ofReal ε := by |
| obtain ⟨U, hxU, hU, hm⟩ := isCompact_singleton.exists_isOpen_lt_of_lt |
| (μ := baseMeasure) (K := {x}) (ENNReal.ofReal ε) (by |
| have hz0 : circleMeasure {x 0} = 0 := by |
| change volume {x 0} = 0 |
| rw [← Metric.closedBall_zero, AddCircle.volume_closedBall] |
| norm_num |
| have hz : baseMeasure {x} = 0 := by |
| apply le_antisymm _ (by positivity) |
| calc |
| baseMeasure {x} ≤ baseMeasure ((fun y : Base => y 0) ⁻¹' {x 0}) := by |
| apply measure_mono |
| intro y hy |
| change y 0 = x 0 |
| exact congrArg (fun z : Base => z 0) (Set.mem_singleton_iff.mp hy) |
| _ = circleMeasure {x 0} := by |
| exact (measurePreserving_eval_infinitePi (fun _ : ℕ => circleMeasure) 0).measure_preimage |
| (measurableSet_singleton (x 0)).nullMeasurableSet |
| _ = 0 := hz0 |
| rw [hz] |
| exact ENNReal.ofReal_pos.mpr hε) |
| exact ⟨U, hxU (Set.mem_singleton x), hU, hm⟩ |
| choose U hxU hU hm using hsmall |
| obtain ⟨t, ht⟩ := isCompact_univ.elim_finite_subcover U hU (by |
| intro x hx |
| exact mem_iUnion.mpr ⟨x, hxU x⟩) |
| let e := Fintype.equivFin t |
| let s : Fin (Fintype.card t) → Set Base := fun i => U (e.symm i) |
| have hs : ∀ i, MeasurableSet (s i) := fun i => (hU (e.symm i)).measurableSet |
| refine ⟨Fintype.card t, fun i => borelSetProjection θ (s i) (hs i), |
| fun i => borelSetProjection_mem θ (s i) (hs i), |
| fun i => borelSetProjection_isStarProjection θ (s i) (hs i), ?_, ?_⟩ |
| · intro i |
| rw [vectorTrace_borelSetProjection] |
| change (baseMeasure (s i)).toReal < ε |
| exact (ENNReal.toReal_lt_of_lt_ofReal (hm (e.symm i))) |
| · apply borelSetProjection_joint_kernel θ s hs |
| intro x |
| obtain ⟨y, hy, hxy⟩ := mem_iUnion₂.mp (ht (mem_univ x)) |
| exact ⟨e ⟨y, hy⟩, by simpa only [s, Equiv.symm_apply_apply] using hxy⟩ |
|
|
| theorem factor_isTypeII1FactorWithVector (θ : ℕ → ℝ) (hθ : JointlyRationallyIndependent θ) : |
| IsTypeII1FactorWithVector (factor θ) traceVector := |
| ⟨factor_isFactor θ hθ, factor_isDiffuse θ, traceVector_faithfulTracial θ⟩ |
|
|
| end Invariant233.ProductModel |
|
|
| end |
|
|
| end OAI |
|
|
