| import Mathlib |
| import OAI.Analysis.Commutator.ConjugatedFreeness |
| import OAI.Analysis.Commutator.FreeMomentsMonotone |
| import OAI.Analysis.Commutator.GeneratedMatrixBicommutant |
| import OAI.Analysis.Commutator.MatrixDiffuse |
| import OAI.Analysis.Commutator.RelativeSequenceRange |
| import OAI.Analysis.Commutator.TraceFaithfulHomInjective |
| import OAI.Analysis.Commutator.TracialFreePositionCriterion |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| attribute [-instance] NonUnitalStarSubalgebra.nonUnitalCStarAlgebra |
|
|
| open scoped InnerProductSpace ComplexOrder Matrix.Norms.L2Operator |
|
|
| open Filter |
|
|
| namespace Kadison.TracialSequence |
|
|
| open FinitePatching.ConjugatedWords |
|
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| variable {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
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| variable (R : VonNeumannAlgebra H) (j : R →⋆ₐ[ℂ] A) |
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| variable (τ : A →ₚ[ℂ] ℂ) (ω : Ultrafilter ℕ) |
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| variable (size : ℕ → ℕ) |
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| variable (stage : ∀ n,Matrix (Fin (size n+1)) (Fin (size n+1)) ℂ →⋆ₐ[ℂ] A) |
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| variable (inc : ∀ {n m : ℕ},n ≤ m → Matrix (Fin (size n+1)) (Fin (size n+1)) ℂ →⋆ₐ[ℂ] |
| Matrix (Fin (size m+1)) (Fin (size m+1)) ℂ) |
|
|
| theorem freeCornerPosition_of_unitary_abstract |
| (hτ : ∀ a b : A,τ (a*b)=τ (b*a)) (hω : (ω : Filter ℕ)≤atTop) |
| (hconstantclosed : Set.centralizer (Set.centralizer (Set.range (constantAction τ ω)))= |
| Set.range (constantAction τ ω)) |
| (hfaith : ∀ a : A,τ (star a*a)=0 → a=0) (hone : τ 1=1) |
| (hirr : ∀ b : A,(∀ n (a : Matrix (Fin (size n+1)) (Fin (size n+1)) ℂ), |
| stage n a*b=b*stage n a) → ∃ c : ℂ,b=algebraMap ℂ A c) |
| (hstage : ∀ n a,stage n a∈Set.range j) |
| (hinc : ∀ n m (h : n ≤ m) a,stage m (inc h a)=stage n a) |
| (hsize : ∀ n q : ℕ,∃ m,n ≤ m ∧ q ≤ size m+1) |
| (U : unitary (boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)) |
| (hu : (U:boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)∈ |
| Set.range ((leftRestricted τ ω).comp (mapSequence j))) |
| (hfree : FreeMoments.Free (pair (A := A) (B := boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω) (constantAction τ ω) U) (sequenceTrace τ ω).toLinearMap) : |
| Nonempty (FreeCornerPosition τ ω R j) := by |
| change Nonempty (FreeCornerPositionData (sequenceVacuum τ ω) (constantAction τ ω) |
| (Set.range ((leftRestricted τ ω).comp (mapSequence j)))) |
| classical |
| let : Algebra ℂ (boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω) := |
| ContinuousLinearMap.algebra |
| let α : (boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω) ≃⋆ₐ[ℂ] |
| (boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω) := |
| (Unitary.linearIsometryEquiv U).conjStarAlgEquiv |
| let φ : A →⋆ₐ[ℂ] (boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω) := by |
| with_reducible |
| exact α.toStarAlgHom.comp (constantAction τ ω) |
| let D := GeneratedMatrixBicommutant.algebra size stage φ |
| let ψ := GeneratedMatrixBicommutant.stageHom size stage φ |
| have hclosed : Set.centralizer (Set.centralizer (Set.range φ))=Set.range φ := by |
| exact EquivariantBicommutant.bicommutant_range α.toMulEquiv (constantAction τ ω) |
| hconstantclosed |
| have hconstantinj : Function.Injective (constantAction τ ω) := by |
| with_reducible |
| refine traceFaithfulHom_injective (A := A) |
| (B := boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω) (constantAction τ ω) |
| τ.toLinearMap (sequenceTrace τ ω).toLinearMap ?_ ?_ |
| · exact hfaith |
| · intro a |
| change sequenceTrace τ ω (leftRestricted τ ω (constantSequence a))=τ a |
| exact trace_constant τ ω a |
| have hφinj : Function.Injective φ := α.injective.comp hconstantinj |
| have hfactor : FactorCorner.IsFactor D := |
| GeneratedMatrixBicommutant.factor size stage φ hclosed hφinj hirr |
| let N := relativeUltrapower τ ω j hτ hω |
| have hUN : (U:boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)∈N := hu |
| have hstageN (n : ℕ) (a : Matrix (Fin (size n+1)) (Fin (size n+1)) ℂ) : φ (stage n a)∈N := by |
| have hc : constantAction τ ω (stage n a)∈N := by |
| let q := Classical.choose (hstage n a) |
| have hq := Classical.choose_spec (hstage n a) |
| refine ⟨constantSequence q,?_⟩ |
| change constantAction τ ω (j q) = constantAction τ ω (stage n a) |
| exact congrArg (constantAction τ ω) hq |
| change (U:boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)* |
| constantAction τ ω (stage n a)*star (U:boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)∈N |
| exact N.mul_mem (N.mul_mem hUN hc) (N.star_mem' hUN) |
| have hDN : D≤N := GeneratedMatrixBicommutant.le_vonNeumann size stage φ N hstageN |
| have hrange (a : D) : (a:boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)∈ |
| Set.range ((leftRestricted τ ω).comp (mapSequence j)) := |
| hDN a.property |
| have hnatural (a : D) : (a:boundedSubspace τ ω →L[ℂ] boundedSubspace τ ω)∈ |
| Set.centralizer (Set.centralizer (Set.range (leftRestricted τ ω))) := by |
| let v := Classical.choose (hrange a) |
| have hv := Classical.choose_spec (hrange a) |
| exact Set.subset_centralizer_centralizer ⟨mapSequence j v,hv⟩ |
| have hfreeD : FreeMoments.Free (GeneratedFreePair.ambient D (constantAction τ ω)) |
| (sequenceTrace τ ω).toLinearMap := by |
| apply hfree.mono |
| intro c |
| cases c with |
| | false => |
| intro a ha |
| exact GeneratedMatrixBicommutant.in_range size stage φ hclosed ⟨a,ha⟩ |
| | true => exact le_refl _ |
| have hdiffuse : DiffuseProjection.HasNoMinimalProjections D := by |
| apply MatrixDiffuse.noMinimal_of_matrix_stages D (sequenceTrace τ ω) |
| (fun a ha b hb => bicommutant_trace_mul τ ω hτ hω (hnatural ⟨a,ha⟩) (hnatural ⟨b,hb⟩)) |
| ((trace_one τ ω).trans hone) hfactor |
| (fun a ha hz => trace_faithful τ ω hτ (hnatural ⟨a,ha⟩) hz) |
| (fun n => size n+1) ψ |
| intro q |
| obtain ⟨m,_,hm⟩ := hsize 0 (q+1) |
| exact ⟨m,by omega⟩ |
| refine ⟨{ algebra := D |
| size := size |
| stage := ψ |
| inclusion := inc |
| compatible := ?_ |
| unbounded := hsize |
| generated := ?_ |
| free := hfreeD |
| in_range := hrange |
| diffuse := hdiffuse }⟩ |
| · intro n m h a |
| apply Subtype.ext |
| change φ (stage m (inc h a))=φ (stage n a) |
| rw [hinc] |
| · exact VectorGNS.generatedByStages_of_represented D (sequenceVacuum τ ω) |
| (fun n => Fin (size n+1)) ψ (GeneratedMatrixBicommutant.in_represented size stage φ) |
| |
| end Kadison.TracialSequence |
| |
| end |
| |
| end OAI |
| |
