| import OAI.Analysis.IrrationalRotation.Model.Spectral |
| import OAI.Analysis.IrrationalRotation.Model.OperatorProperties |
| import OAI.Analysis.IrrationalRotation.Model.GeneratedVonNeumann |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped ENNReal |
| namespace Invariant233 |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| def vectorFugledeKadisonDeterminant (ξ : H) (T : H →L[ℂ] H) : ℝ≥0∞ := |
| ⨅ ε : {r : ℝ // 0 < r}, ENNReal.ofReal |
| (Real.exp ((vectorTrace ξ (CFC.log (star T * T + ε.val • (1 : H →L[ℂ] H)))).re / 2)) |
|
|
| def IsVectorBrownMeasure (ξ : H) (T : H →L[ℂ] H) (μ : Measure ℂ) : Prop := |
| IsProbabilityMeasure μ ∧ IsCompact μ.support ∧ ∀ z : ℂ, |
| vectorFugledeKadisonDeterminant ξ (T - z • (1 : H →L[ℂ] H)) = |
| logarithmicPotentialDeterminant μ z |
|
|
| def operatorModulus (T : H →L[ℂ] H) : H →L[ℂ] H := CFC.sqrt (star T * T) |
|
|
| def HasCountablyAtomicSpectralType (B : H →L[ℂ] H) : Prop := |
| ∃ S : Set ℂ, S.Countable ∧ |
| (⨆ z : S, Module.End.eigenspace B.toLinearMap z.val).topologicalClosure = ⊤ |
|
|
| def polarConjugate (U B : H →L[ℂ] H) (n : ℤ) : H →L[ℂ] H := |
| if 0 ≤ n then U ^ n.toNat * B * star U ^ n.toNat |
| else star U ^ (-n).toNat * B * U ^ (-n).toNat |
|
|
| def finitePolarAlgebra (U B : H →L[ℂ] H) (F : Finset ℤ) : VonNeumannAlgebra H := |
| vonNeumannGenerated {A | ∃ n ∈ F, A = polarConjugate U B n} |
|
|
| def fullPolarAlgebra (U B : H →L[ℂ] H) : VonNeumannAlgebra H := |
| vonNeumannGenerated (Set.range (polarConjugate U B)) |
|
|
| def ProductPolarStructure (U T : H →L[ℂ] H) : Prop := |
| star U * U = 1 ∧ U * star U = 1 ∧ T = U * operatorModulus T ∧ |
| Function.Injective (operatorModulus T) ∧ HasCountablyAtomicSpectralType (operatorModulus T) ∧ |
| (∀ F : Finset ℤ, |
| IsAbelianVonNeumannAlgebra (finitePolarAlgebra U (operatorModulus T) F) ∧ |
| IsAtomicVonNeumannAlgebra (finitePolarAlgebra U (operatorModulus T) F)) ∧ |
| IsAbelianVonNeumannAlgebra (fullPolarAlgebra U (operatorModulus T)) ∧ |
| IsDiffuseVonNeumannAlgebra (fullPolarAlgebra U (operatorModulus T)) |
|
|
| end Invariant233 |
|
|
| end |
|
|
| end OAI |
|
|
