| import Mathlib |
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| namespace OAI |
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| noncomputable section |
|
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| namespace FreeGroupFactorMain |
|
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| universe u uG |
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| open scoped ENNReal |
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|
| abbrev GroupHilbert (G : Type u) := lp (fun _ : G => ℂ) 2 |
| abbrev GroupOperator (G : Type u) := GroupHilbert G →L[ℂ] GroupHilbert G |
|
|
| def leftTranslations (G : Type u) [Group G] : Set (GroupOperator G) := |
| {T | ∃ g : G, ∀ (ξ : GroupHilbert G) (h : G), (T ξ) h = ξ (g⁻¹ * h)} |
|
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| def groupVonNeumann (G : Type u) [Group G] : StarSubalgebra ℂ (GroupOperator G) := |
| StarSubalgebra.centralizer ℂ |
| (StarSubalgebra.centralizer ℂ (leftTranslations G) : Set (GroupOperator G)) |
|
|
| def canonicalTrace {G : Type u} [Group G] (T : groupVonNeumann G) : ℂ := by |
| classical |
| exact ((T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))) 1 |
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|
| @[instance_reducible] def ultraweakOperatorTopology (G : Type u) : TopologicalSpace (GroupOperator G) := |
| ⨅ (ξ : lp (fun _ : ℕ => GroupHilbert G) 2) |
| (η : lp (fun _ : ℕ => GroupHilbert G) 2), |
| TopologicalSpace.induced |
| (fun T : GroupOperator G => ∑' k : ℕ, inner ℂ (η k) (T (ξ k))) |
| inferInstance |
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| @[instance_reducible] def groupUltraweakTopology (G : Type u) [Group G] : |
| TopologicalSpace (groupVonNeumann G) := |
| TopologicalSpace.induced (fun T : groupVonNeumann G => (T : GroupOperator G)) |
| (ultraweakOperatorTopology G) |
|
|
| abbrev FreeGroupRank (n : ℕ) := FreeGroup (Fin n) |
|
|
| instance groupVN_isClosed (G : Type uG) [Group G] : |
| IsClosed (groupVonNeumann G : Set (GroupOperator G)) := by |
| unfold groupVonNeumann |
| rw [StarSubalgebra.coe_centralizer] |
| exact Set.isClosed_centralizer _ |
|
|
| instance groupVN_CStarAlgebra (G : Type uG) [Group G] : |
| CStarAlgebra (groupVonNeumann G) := inferInstance |
|
|
| end FreeGroupFactorMain |
|
|
| universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_42 u_43 |
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| namespace FreeGroupFactorMain.Amplification |
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| open scoped ENNReal |
|
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| section |
| variable {A : Type u_1} {B : Type u_2} [CStarAlgebra A] [CStarAlgebra B] |
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| def cornerSubalgebra (p : A) (hp : IsStarProjection p) : NonUnitalStarSubalgebra ℂ A where |
| carrier := {x | p*x=x ∧ x*p=x} |
| zero_mem' := by simp |
| add_mem' hx hy := by constructor <;> simp [mul_add,add_mul,hx.1,hx.2,hy.1,hy.2] |
| mul_mem' hx hy := by |
| constructor |
| · rw [← mul_assoc,hx.1] |
| · rw [mul_assoc,hy.2] |
| smul_mem' c x hx := by |
| constructor |
| · rw [mul_smul_comm,hx.1] |
| · rw [smul_mul_assoc,hx.2] |
| star_mem' hx := by |
| constructor |
| · simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hx.2 |
| · simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hx.1 |
|
|
| abbrev Corner (p : A) (hp : IsStarProjection p) := ↥(cornerSubalgebra p hp) |
|
|
| instance cornerClosed (p : A) (hp : IsStarProjection p) : |
| IsClosed (cornerSubalgebra p hp : Set A) := |
| (isClosed_eq (continuous_const.mul continuous_id) continuous_id).inter |
| (isClosed_eq (continuous_id.mul continuous_const) continuous_id) |
|
|
| instance cornerSemiring (p : A) (hp : IsStarProjection p) : Semiring (Corner p hp) where |
| __ : NonUnitalSemiring (Corner p hp) := inferInstance |
| one := ⟨p,hp.isIdempotentElem.eq,hp.isIdempotentElem.eq⟩ |
| one_mul x := Subtype.ext x.property.1 |
| mul_one x := Subtype.ext x.property.2 |
|
|
| instance cornerRing (p : A) (hp : IsStarProjection p) : Ring (Corner p hp) where |
| __ : Semiring (Corner p hp) := inferInstance |
| __ : NonUnitalRing (Corner p hp) := inferInstance |
|
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| instance cornerAlgebra (p : A) (hp : IsStarProjection p) : Algebra ℂ (Corner p hp) := |
| Algebra.ofModule smul_mul_assoc mul_smul_comm |
|
|
| instance cornerNormedRing (p : A) (hp : IsStarProjection p) : NormedRing (Corner p hp) where |
| __ : Ring (Corner p hp) := inferInstance |
| __ : NonUnitalNormedRing (Corner p hp) := inferInstance |
|
|
| instance cornerCStarAlgebra (p : A) (hp : IsStarProjection p) : CStarAlgebra (Corner p hp) where |
| __ : NonUnitalCStarAlgebra (Corner p hp) := inferInstance |
| __ : NormedRing (Corner p hp) := inferInstance |
| __ : Algebra ℂ (Corner p hp) := inferInstance |
| norm_smul_le := norm_smul_le |
|
|
| end |
|
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| section |
| variable {A : Type u_3} {B : Type u_4} [CStarAlgebra A] [CStarAlgebra B] |
|
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| @[instance_reducible] def cornerTopology (𝒯 : TopologicalSpace A) |
| (p : A) (hp : IsStarProjection p) : TopologicalSpace (Corner p hp) := |
| TopologicalSpace.induced (fun x : Corner p hp => (x : A)) 𝒯 |
|
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| def normalizedCornerTrace (τ : A → ℂ) (t : ℝ) (p : A) (hp : IsStarProjection p) |
| (x : Corner p hp) : ℂ := (t : ℂ)⁻¹ * τ x |
|
|
| end |
|
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| def stabilizedRightTranslations (G : Type u_5) [Group G] : Set (GroupOperator (ℕ × G)) := |
| {T | ∃ g : G, ∀ (ξ : GroupHilbert (ℕ × G)) (x : ℕ × G), |
| T ξ x = ξ (x.1,x.2*g)} |
|
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| def stabilization (G : Type u_6) [Group G] : StarSubalgebra ℂ (GroupOperator (ℕ × G)) := |
| StarSubalgebra.centralizer ℂ (stabilizedRightTranslations G) |
|
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| instance stabilization_isClosed (G : Type u_7) [Group G] : |
| IsClosed (stabilization G : Set (GroupOperator (ℕ × G))) := by |
| rw [stabilization,StarSubalgebra.coe_centralizer] |
| exact Set.isClosed_centralizer _ |
|
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| instance stabilization_CStarAlgebra (G : Type u_8) [Group G] : CStarAlgebra (stabilization G) := |
| inferInstance |
|
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| def stabilizedProjectionTrace {G : Type u_9} [Group G] (p : stabilization G) : ℝ≥0∞ := by |
| classical |
| exact ∑' k : ℕ, ENNReal.ofReal (‖(p : GroupOperator (ℕ × G)) |
| (lp.single 2 (k,1) (1 : ℂ))‖ ^ 2) |
|
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| def stabilizedTrace {G : Type u_10} [Group G] (T : stabilization G) : ℂ := by |
| classical |
| exact ∑' k : ℕ, inner ℂ (lp.single 2 (k,1) (1 : ℂ)) |
| ((T : GroupOperator (ℕ × G)) (lp.single 2 (k,1) (1 : ℂ))) |
|
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| @[instance_reducible] def stabilizationTopology (G : Type u_11) [Group G] : |
| TopologicalSpace (stabilization G) := |
| TopologicalSpace.induced (fun T : stabilization G => (T : GroupOperator (ℕ × G))) |
| (ultraweakOperatorTopology (ℕ × G)) |
|
|
| section |
| variable {A : Type u_42} {B : Type u_43} [CStarAlgebra A] [CStarAlgebra B] |
|
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| structure NormalTracialEquiv (τA : A → ℂ) (τB : B → ℂ) |
| (𝒯A : TopologicalSpace A) (𝒯B : TopologicalSpace B) where |
| equiv : A ≃⋆ₐ[ℂ] B |
| trace_eq : ∀x,τB (equiv x)=τA x |
| normal : @Continuous _ _ 𝒯A 𝒯B equiv |
| inverse_normal : @Continuous _ _ 𝒯B 𝒯A equiv.symm |
|
|
| end |
|
|
| end FreeGroupFactorMain.Amplification |
|
|
| namespace FreeGroupFactorMain.Interpolation |
|
|
| open Amplification |
| open scoped ENNReal |
|
|
| abbrev RankTwoProjection := {p : stabilization (FreeGroupRank 2) // IsStarProjection p} |
|
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| instance rankTwoProjectionInhabited : Inhabited RankTwoProjection := |
| ⟨⟨0,IsStarProjection.zero _⟩⟩ |
|
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| def selectedProjection (a : ℝ) : RankTwoProjection := |
| Classical.epsilon (fun p : RankTwoProjection => stabilizedProjectionTrace p.val=ENNReal.ofReal a) |
|
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| def interpolationScale (r : ℝ) : ℝ := (Real.sqrt (r-1))⁻¹ |
|
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| inductive Model where |
| | integer (n : ℕ) |
| | continuous (r : ℝ) |
| | infinity |
|
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| def Model.Algebra : Model → Type |
| | .integer n => groupVonNeumann (FreeGroupRank n) |
| | .continuous r => Corner (selectedProjection (interpolationScale r)).val |
| (selectedProjection (interpolationScale r)).property |
| | .infinity => groupVonNeumann (FreeGroup ℕ) |
|
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| instance modelCStarAlgebra (m : Model) : CStarAlgebra m.Algebra := by |
| cases m <;> dsimp only [Model.Algebra] <;> infer_instance |
|
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| def Model.trace : (m : Model) → m.Algebra → ℂ |
| | .integer _ => canonicalTrace |
| | .continuous r => normalizedCornerTrace stabilizedTrace (interpolationScale r) |
| (selectedProjection (interpolationScale r)).val (selectedProjection (interpolationScale r)).property |
| | .infinity => canonicalTrace |
|
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| @[instance_reducible] def Model.topology : (m : Model) → TopologicalSpace m.Algebra |
| | .integer n => groupUltraweakTopology (FreeGroupRank n) |
| | .continuous r => cornerTopology (stabilizationTopology (FreeGroupRank 2)) |
| (selectedProjection (interpolationScale r)).val (selectedProjection (interpolationScale r)).property |
| | .infinity => groupUltraweakTopology (FreeGroup ℕ) |
|
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| def parameterModel (r : ℝ≥0∞) : Model := by |
| classical |
| exact if r=⊤ then .infinity else |
| if h : ∃n : ℕ,r=(n:ℝ≥0∞) then .integer h.choose else .continuous r.toReal |
|
|
| abbrev InterpolatedFactor (r : ℝ≥0∞) := (parameterModel r).Algebra |
| abbrev interpolatedTrace (r : ℝ≥0∞) := (parameterModel r).trace |
| abbrev interpolatedTopology (r : ℝ≥0∞) := (parameterModel r).topology |
|
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| end FreeGroupFactorMain.Interpolation |
|
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| namespace FreeGroupFactorMain |
|
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| universe u v |
|
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| def HasNormalTracialIso (G : Type u) (H : Type v) [Group G] [Group H] : Prop := |
| ∃ Φ : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann H, |
| (∀ T, canonicalTrace (Φ T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology G) (groupUltraweakTopology H) Φ |
|
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| end FreeGroupFactorMain |
|
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| end |
|
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| end OAI |
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|
