| import Mathlib |
| import OAI.Analysis.FiniteFactor.CrossedProduct.VacuumFunctional |
|
|
| namespace OAI |
|
|
| section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology ENNReal CStarAlgebra |
| noncomputable section |
| namespace FiniteFactor |
| local instance : IsometricContinuousFunctionalCalculus ℝ |
| (SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) IsSelfAdjoint := |
| IsSelfAdjoint.instIsometricContinuousFunctionalCalculus |
| instance productHilbertNontrivial : Nontrivial (SourceHilbert SourceCoordinate) := by |
| have hv : sourceVacuum≠0 := by |
| intro hz |
| have h := sourceVacuum_norm |
| rw [hz,norm_zero] at h |
| norm_num at h |
| exact ⟨⟨sourceVacuum,0,hv⟩⟩ |
| |
| end FiniteFactor |
| end |
| end |
| |
| section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology |
| noncomputable section |
| namespace FiniteFactor |
| local notation "productMeasurable" => |
| @Prod.instMeasurableSpace (SourceCoordinate → ℝ) ℤ |
| (@MeasurableSpace.pi SourceCoordinate (fun _ => ℝ) |
| (fun _ => Real.measurableSpace)) Int.instMeasurableSpace |
| local notation "productSemiring" => |
| Complex.instNormedField.toField.toCommRing.toRing.toSemiring |
| local notation "productIdentity" => |
| @RingHom.id ℂ (@Semiring.toNonAssocSemiring ℂ productSemiring) |
| local notation "productExponent" => |
| @Nat.cast ENNReal |
| (ENNReal.instAddCommMonoidWithOne.toAddMonoidWithOne.toNatCast) |
| (Nat.succ (Nat.succ Nat.zero)) |
| local notation "productHilbertGroup" => |
| @MeasureTheory.Lp.instNormedAddCommGroup |
| ((SourceCoordinate → ℝ) × ℤ) ℂ productMeasurable |
| productExponent |
| (sourceMeasure SourceCoordinate) Complex.instNormedAddCommGroup |
| fact_one_le_two_ennreal |
| local notation "productHilbertInner" => |
| @MeasureTheory.L2.innerProductSpace |
| ((SourceCoordinate → ℝ) × ℤ) ℂ ℂ Complex.instRCLike |
| productMeasurable (sourceMeasure SourceCoordinate) |
| Complex.instNormedAddCommGroup (@RCLike.innerProductSpace ℂ Complex.instRCLike) |
| local notation "productHilbertTopology" => |
| @UniformSpace.toTopologicalSpace (SourceHilbert SourceCoordinate) |
| (@PseudoMetricSpace.toUniformSpace (SourceHilbert SourceCoordinate) |
| (@MetricSpace.toPseudoMetricSpace (SourceHilbert SourceCoordinate) |
| (@NormedAddCommGroup.toMetricSpace (SourceHilbert SourceCoordinate) productHilbertGroup))) |
| local notation "productHilbertAddition" => |
| @AddCommGroup.toAddCommMonoid (SourceHilbert SourceCoordinate) |
| (@NormedAddCommGroup.toAddCommGroup (SourceHilbert SourceCoordinate) productHilbertGroup) |
| local notation "productHilbertModule" => |
| @NormedSpace.toModule ℂ (SourceHilbert SourceCoordinate) |
| (@DenselyNormedField.toNormedField ℂ |
| (@RCLike.toDenselyNormedField ℂ Complex.instRCLike)) |
| (@NormedAddCommGroup.toSeminormedAddCommGroup |
| (SourceHilbert SourceCoordinate) productHilbertGroup) |
| (@InnerProductSpace.toNormedSpace ℂ (SourceHilbert SourceCoordinate) Complex.instRCLike |
| (@NormedAddCommGroup.toSeminormedAddCommGroup |
| (SourceHilbert SourceCoordinate) productHilbertGroup) |
| productHilbertInner) |
| |
| abbrev ProductOp := |
| @ContinuousLinearMap ℂ ℂ productSemiring productSemiring productIdentity |
| (SourceHilbert SourceCoordinate) productHilbertTopology productHilbertAddition |
| (SourceHilbert SourceCoordinate) productHilbertTopology productHilbertAddition |
| productHilbertModule productHilbertModule |
| |
| section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology ENNReal |
| local instance : Module ℝ ProductOp := Module.complexToReal ProductOp |
| local instance : IsometricContinuousFunctionalCalculus ℝ ProductOp IsSelfAdjoint := |
| IsSelfAdjoint.instIsometricContinuousFunctionalCalculus |
| |
| def logarithmCutoff (n : ℕ) : C(ℝ,ℝ) where |
| toFun t := Real.log (max (Real.exp (-(n:ℝ))) t) |
| continuous_toFun := (continuous_const.max continuous_id).log |
| (fun _=>ne_of_gt ((Real.exp_pos _).trans_le (le_max_left _ _))) |
|
|
| run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {} |
|
|
| def FugledeKadisonDeterminant (τ : ProductOp → ℂ) (T : ProductOp) : ℝ≥0∞ := |
| let determinant := fun (Hilbert : Type) [NormedAddCommGroup Hilbert] |
| [InnerProductSpace ℂ Hilbert] [CompleteSpace Hilbert] [Nontrivial Hilbert] |
| (trace : (Hilbert →L[ℂ] Hilbert) → ℂ) (operator : Hilbert →L[ℂ] Hilbert) => |
| letI : Module ℝ (Hilbert →L[ℂ] Hilbert) := Module.complexToReal _ |
| letI : IsometricContinuousFunctionalCalculus ℝ (Hilbert →L[ℂ] Hilbert) |
| IsSelfAdjoint := IsSelfAdjoint.instIsometricContinuousFunctionalCalculus |
| ⨅ n : ℕ, ENNReal.ofReal |
| (Real.exp ((trace (cfc (logarithmCutoff n) (CFC.abs operator))).re)) |
| let domain := (SourceCoordinate → ℝ) × ℤ |
| let measurable : MeasurableSpace domain := productMeasurable |
| let exponent : ℝ≥0∞ := productExponent |
| let measure : @Measure domain measurable := sourceMeasure SourceCoordinate |
| let complete := @MeasureTheory.Lp.instCompleteSpace domain measurable exponent measure ℂ |
| Complex.instNormedAddCommGroup Complex.instCompleteSpace fact_one_le_two_ennreal |
| @determinant (SourceHilbert SourceCoordinate) productHilbertGroup productHilbertInner |
| complete productHilbertNontrivial τ T |
|
|
| end |
|
|
| section |
| open MeasureTheory Filter |
| open scoped ENNReal Topology |
|
|
| def expLogPotential (μ : Measure ℂ) (z : ℂ) : ℝ≥0∞ := |
| ⨅ n : ℕ, ENNReal.ofReal (Real.exp (∫ w,logarithmCutoff n ‖z-w‖ ∂μ)) |
|
|
| def IsBrownMeasure (τ : ProductOp → ℂ) (T : ProductOp) (μ : Measure ℂ) : Prop := |
| let translate := fun (Hilbert : Type) [NormedAddCommGroup Hilbert] |
| [NormedSpace ℂ Hilbert] (operator : Hilbert →L[ℂ] Hilbert) (scalar : ℂ) => |
| operator - algebraMap ℂ (Hilbert →L[ℂ] Hilbert) scalar |
| IsProbabilityMeasure μ ∧ IsCompact μ.support ∧ |
| ∀ z : ℂ, FugledeKadisonDeterminant τ |
| (@translate (SourceHilbert SourceCoordinate) productHilbertGroup |
| (@InnerProductSpace.toNormedSpace ℂ (SourceHilbert SourceCoordinate) Complex.instRCLike |
| (@NormedAddCommGroup.toSeminormedAddCommGroup |
| (SourceHilbert SourceCoordinate) productHilbertGroup) |
| productHilbertInner) T z) = expLogPotential μ z |
|
|
| def ProductBrownClaim : Prop := |
| ∀ (θ : SourceCoordinate → ℝ) (hθ : SourceAngles θ), |
| IsBrownMeasure sourceVacuumFunctional |
| (sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg) (Measure.dirac 0) |
|
|
| end |
|
|
| end FiniteFactor |
| end |
| end |
|
|
| end OAI |
|
|
