| import OAI.Analysis.FreeGroupFactor.FiniteMoments |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| universe uA uι uκ |
|
|
| open Set Metric Filter |
| open scoped NNReal Topology BigOperators ContDiff ENNReal |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| namespace FreeGroupFactorMain.FiniteActive |
|
|
| section |
| open UnitaryEvaluation |
| open scoped ENNReal |
| theorem original_generators_mem {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ] |
| (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (X : ℝ → ι → Ambient ι κ) (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ)) |
| (hX₀ : X 0 = fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) |
| (hX : ∀ t, HasDerivAt X (PolynomialFlow.field |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) |
| (E : StarSubalgebra ℂ (Ambient ι κ)) (hE : IsClosed (E : Set (Ambient ι κ))) |
| (hC : (generatorUnitary (none : Option (ι ⊕ κ)) : Ambient ι κ) ∈ E) |
| (t₀ : ℝ) (hXt : ∀ i, X t₀ i ∈ E) |
| (hpass : ∀ k : κ, (generatorUnitary (some (Sum.inr k) : Option (ι ⊕ κ)) : Ambient ι κ) ∈ E) : |
| ∀ i : Option (ι ⊕ κ), (generatorUnitary i : Ambient ι κ) ∈ E := by |
| have hXS := PolynomialFlow.solution_mem_subalgebra E hE |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) hS |
| (laurentValue_mem E _ hC l) h X hX hU t₀ hXt |
| intro i |
| cases i with |
| | none => exact hC |
| | some j => |
| cases j with |
| | inl i => |
| change groupUnitary (FreeGroup.of (some (Sum.inl i))) ∈ E |
| simpa only [hX₀] using hXS 0 i |
| | inr k => exact hpass k |
|
|
| theorem orbit_total {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ] |
| (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (X : ℝ → ι → groupVonNeumann (FreeGroup (Option (ι ⊕ κ)))) |
| (hU : ∀ t j, X t j ∈ unitary (groupVonNeumann (FreeGroup (Option (ι ⊕ κ))))) |
| (hX₀ : X 0 = fun j => groupUnitary (FreeGroup.of (some (Sum.inl j)))) |
| (hX : ∀ t, HasDerivAt X (PolynomialFlow.field |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) (t₀ : ℝ) : |
| ⊤ ≤ (Submodule.span ℂ (Set.range (fun g : FreeGroup (Option (ι ⊕ κ)) => |
| ((wordHom (extendUnitary (fun j => ⟨tuple (X t₀) j,tuple_unitary (X t₀) (hU t₀) j⟩) |
| (generatorUnitary none)) g : groupVonNeumann (FreeGroup (Option (ι ⊕ κ)))) : |
| GroupOperator (FreeGroup (Option (ι ⊕ κ)))) deltaOne))).topologicalClosure := by |
| classical |
| let U := extendUnitary (fun j => ⟨tuple (X t₀) j,tuple_unitary (X t₀) (hU t₀) j⟩) (generatorUnitary none) |
| let ρ : FreeGroup (Option (ι ⊕ κ)) →* GroupOperator (FreeGroup (Option (ι ⊕ κ))) := |
| (groupVonNeumann _).subtype.toMonoidHom.comp (wordHom U) |
| let K := (Submodule.span ℂ (Set.range (fun g => ρ g deltaOne))).topologicalClosure |
| let E := (reducingAlgebra K).comap (groupVonNeumann _).subtype |
| have hK : IsClosed (K : Set (GroupHilbert (FreeGroup (Option (ι ⊕ κ))))) := |
| Submodule.isClosed_topologicalClosure _ |
| have hE : IsClosed (E : Set (groupVonNeumann (FreeGroup (Option (ι ⊕ κ))))) := |
| (reducingAlgebra_isClosed K hK).preimage continuous_subtype_val |
| have hρ (g) : wordHom U g ∈ E := by |
| constructor |
| · exact closedSpan_invariant ρ deltaOne g |
| · have he : star (ρ g) = ρ g⁻¹ := by |
| change ((star (wordHom U g) : groupVonNeumann (FreeGroup (Option (ι ⊕ κ)))) : |
| GroupOperator (FreeGroup (Option (ι ⊕ κ)))) = _ |
| rw [← wordHom_inv] |
| rfl |
| change ∀ x ∈ K, (star (ρ g)) x ∈ K |
| rw [he] |
| exact closedSpan_invariant ρ deltaOne g⁻¹ |
| have hC : (generatorUnitary (none : Option (ι ⊕ κ)) : groupVonNeumann _) ∈ E := by |
| simpa only [wordHom_of, U, extendUnitary, Option.elim_none] using hρ (FreeGroup.of none) |
| have hXt : ∀ j, X t₀ j ∈ E := by |
| intro j |
| simpa only [wordHom_of, U, extendUnitary, Option.elim_some,tuple,Sum.elim_inl] using hρ (FreeGroup.of (some (Sum.inl j))) |
| have hpass (k : κ) : (generatorUnitary (some (Sum.inr k) : Option (ι ⊕ κ)) : Ambient ι κ) ∈ E := by |
| change groupUnitary (FreeGroup.of (some (Sum.inr k))) ∈ E |
| simpa only [wordHom_of,U,extendUnitary,Option.elim_some,tuple,Sum.elim_inr] |
| using hρ (FreeGroup.of (some (Sum.inr k))) |
| have hgen := original_generators_mem h l hS X hU hX₀ hX E hE hC t₀ hXt hpass |
| change ⊤ ≤ K |
| apply canonical_total_of_reducing K hK _ hgen |
| apply Submodule.le_topologicalClosure |
| apply Submodule.subset_span |
| exact ⟨1, by simp⟩ |
|
|
| end |
| open UnitaryEvaluation |
|
|
| theorem extension_at_time {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ] |
| (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (X : ℝ → ι → Ambient ι κ) (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ)) |
| (hX₀ : X 0 = fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) |
| (hX : ∀ t, HasDerivAt X (PolynomialFlow.field |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) (t : ℝ) : |
| ∃ Φ : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ, |
| (∀ T, canonicalTrace (Φ T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) |
| (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) Φ ∧ |
| ∀ g, Φ (groupUnitary g) = wordHom |
| (extendUnitary (fun j => ⟨tuple (X t) j,tuple_unitary (X t) (hU t) j⟩) |
| (generatorUnitary none)) g := by |
| exact iso_extension_of_haar_total |
| (wordHom (extendUnitary (fun j => ⟨tuple (X t) j,tuple_unitary (X t) (hU t) j⟩) |
| (generatorUnitary none))) |
| (fun g => (wordHom_inv _ g).symm) |
| (moments h l hS X hU hX₀ hX t) |
| (orbit_total h l hS X hU hX₀ hX t) |
|
|
| theorem basis_solution {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ] |
| (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) : |
| ∃ (X : ℝ → ι → Ambient ι κ), |
| X 0 = (fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) ∧ |
| (∀ t, HasDerivAt X (PolynomialFlow.field |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) ∧ |
| ∀ t i, X t i ∈ unitary (Ambient ι κ) := by |
| exact PolynomialFlow.global_unitary_solution (A := Ambient ι κ) |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) hS h |
| (fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) (fun _ => groupUnitary_mem_unitary _) |
|
|
| theorem automorphisms {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ] |
| (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) : |
| ∃ (X : ℝ → ι → Ambient ι κ) |
| (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ)), |
| X 0 = (fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) ∧ |
| (∀ t, HasDerivAt X (PolynomialFlow.field |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) ∧ |
| ∀ t, ∃ Φ : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ, |
| (∀ T, canonicalTrace (Φ T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) |
| (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) Φ ∧ |
| ∀ g, Φ (groupUnitary g) = wordHom |
| (extendUnitary (fun j => ⟨tuple (X t) j,tuple_unitary (X t) (hU t) j⟩) |
| (generatorUnitary none)) g := by |
| obtain ⟨X,hX₀,hX,hU⟩ := basis_solution (ι := ι) (κ := κ) h l hS |
| refine ⟨X,hU,hX₀,hX,?_⟩ |
| intro t |
| exact extension_at_time h l hS X hU hX₀ hX t |
|
|
| end FreeGroupFactorMain.FiniteActive |
|
|
| namespace FreeGroupFactorMain.UnitaryEvaluation |
| variable {A : Type uA} [CStarAlgebra A] {ι : Type uι} {κ : Type uκ} |
|
|
| theorem hamiltonianLinear_mapDomain (S : A) (U : κ → unitary A) |
| (f : ι → κ) (k : FreeGroup ι →₀ ℝ) : |
| hamiltonianLinear S U (Finsupp.mapDomain (FreeGroup.map f) k) = |
| hamiltonianLinear S (U ∘ f) k := by |
| classical |
| simp only [hamiltonianLinear,Finsupp.linearCombination_apply] |
| rw [Finsupp.sum_mapDomain_index (fun g => by simp) |
| (fun g a b => by simp [add_smul])] |
| apply Finsupp.sum_congr |
| intro g hg |
| rw [wordHom_map] |
|
|
| end FreeGroupFactorMain.UnitaryEvaluation |
|
|
| namespace FreeGroupFactorMain.FiniteActive |
|
|
| section |
| open UnitaryEvaluation |
| variable {ι : Type uι} {κ : Type uκ} [Fintype ι] [DecidableEq ι] [DecidableEq κ] |
|
|
| omit [Fintype ι] in |
| theorem flow_bound (l : ℤ →₀ ℂ) (h0 : l 0 = 0) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (K : ℝ) (hK : 0 ≤ K) |
| (hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K) |
| (U : ι → unitary (Ambient ι κ)) |
| (Φ : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ) |
| (hΦ : ∀ g, Φ (groupUnitary g) = wordHom |
| (extendUnitary (fun j => ⟨tuple (fun i => (U i : Ambient ι κ)) j, |
| tuple_unitary _ (fun i => (U i).property) j⟩) (generatorUnitary none)) g) |
| (k : FreeGroup ι →₀ ℝ) : |
| ‖hamiltonianLinear (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) U k‖ ≤ |
| 3*K*Real.sqrt (k.sum (fun _ r => r^2)) := by |
| let V : (ι ⊕ κ) → unitary (Ambient ι κ) := fun j => |
| ⟨tuple (fun i => (U i : Ambient ι κ)) j,tuple_unitary _ (fun i => (U i).property) j⟩ |
| have hb := flow_iso_hamiltonian_bound l h0 hS K hK hn V Φ hΦ |
| (Finsupp.mapDomain (FreeGroup.map Sum.inl) k) |
| rw [hamiltonianLinear_mapDomain, |
| Finsupp.sum_mapDomain_index_inj (FreeGroup.map_injective Sum.inl_injective)] at hb |
| exact hb |
|
|
| theorem bounded_solution (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) (h0 : l 0 = 0) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (K : ℝ) (hK : 0 ≤ K) |
| (hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K) |
| (X : ℝ → ι → Ambient ι κ) (hU : ∀ t i, X t i ∈ unitary (Ambient ι κ)) |
| (hX₀ : X 0 = fun i => groupUnitary (FreeGroup.of (some (Sum.inl i)))) |
| (hX : ∀ t, HasDerivAt X (PolynomialFlow.field |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h (X t)) t) : |
| ∀ t k, ‖hamiltonianLinear |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) |
| (fun i => ⟨X t i,hU t i⟩) k‖ ≤ 3*K*Real.sqrt (k.sum (fun _ r => r^2)) := by |
| intro t k |
| obtain ⟨Φ,_ht,_hc,hΦ⟩ := extension_at_time (ι := ι) (κ := κ) h l hS X hU hX₀ hX t |
| exact flow_bound (ι := ι) (κ := κ) l h0 hS K hK hn (fun i => ⟨X t i,hU t i⟩) Φ hΦ k |
|
|
| theorem bounded_flow (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) (h0 : l 0 = 0) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (K : ℝ) (hK : 0 ≤ K) |
| (hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K) : |
| ∃ U : ℝ → ι → unitary (Ambient ι κ), |
| U 0 = (fun i => generatorUnitary (some (Sum.inl i) : Option (ι ⊕ κ))) ∧ |
| (∀ t, HasDerivAt (fun t i => (U t i : Ambient ι κ)) |
| (PolynomialFlow.field (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) |
| h (fun i => (U t i : Ambient ι κ))) t) ∧ |
| (∀ t k, ‖hamiltonianLinear |
| (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) (U t) k‖ ≤ |
| 3*K*Real.sqrt (k.sum (fun _ r => r^2))) ∧ |
| ∃ β : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ, |
| (∀ T, canonicalTrace (β T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) |
| (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) β ∧ |
| ∀ g, β (groupUnitary g) = wordHom |
| (extendUnitary (fun j => ⟨tuple (fun i => (U 1 i : Ambient ι κ)) j, |
| tuple_unitary _ (fun i => (U 1 i).property) j⟩) (generatorUnitary none)) g := by |
| obtain ⟨X,hX₀,hX,hU⟩ := basis_solution (ι := ι) (κ := κ) h l hS |
| let U (t : ℝ) : ι → unitary (Ambient ι κ) := fun i => ⟨X t i,hU t i⟩ |
| refine ⟨U,?_,hX,?_,?_⟩ |
| · funext i |
| exact Subtype.ext (congrFun hX₀ i) |
| · exact bounded_solution (ι := ι) (κ := κ) h l h0 hS K hK hn X hU hX₀ hX |
| · exact extension_at_time (ι := ι) (κ := κ) h l hS X hU hX₀ hX 1 |
|
|
| omit [Fintype ι] [DecidableEq ι] [DecidableEq κ] in |
| theorem wordHom_active (w : FreeGroup ι) : |
| wordHom (fun i => generatorUnitary (some (Sum.inl i) : Option (ι ⊕ κ))) w = |
| groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w) := by |
| rw [← wordHom_generatorUnitary,wordHom_map] |
| rfl |
|
|
| omit [Fintype ι] [DecidableEq ι] [DecidableEq κ] in |
| theorem wordHom_tuple_active (U : ι → unitary (Ambient ι κ)) (w : FreeGroup ι) : |
| wordHom (extendUnitary (fun j => ⟨tuple (fun i => (U i : Ambient ι κ)) j, |
| tuple_unitary _ (fun i => (U i).property) j⟩) (generatorUnitary none)) |
| (FreeGroup.map (fun i => some (Sum.inl i)) w) = wordHom U w := by |
| rw [wordHom_map] |
| rfl |
|
|
| omit [DecidableEq κ] in |
| theorem displacements_from_bounds |
| (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) (K : ℝ) |
| (U : ℝ → ι → unitary (Ambient ι κ)) |
| (hU₀ : U 0 = fun j => generatorUnitary (some (Sum.inl j) : Option (ι ⊕ κ))) |
| (hX : ∀ t, HasDerivAt (fun t j => (U t j : Ambient ι κ)) |
| (PolynomialFlow.field (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h |
| (fun j => (U t j : Ambient ι κ))) t) |
| (hbound : ∀ t k, ‖hamiltonianLinear (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) (U t) k‖ ≤ |
| 3*K*Real.sqrt (k.sum (fun _ r => r^2))) : |
| (∀ j, ‖(U 1 j : Ambient ι κ) - groupUnitary (FreeGroup.of (some (Sum.inl j)))‖ ≤ |
| 3*K*Real.sqrt ((h j).sum (fun _ r => r^2))) ∧ |
| ∀ w : FreeGroup ι, ‖wordHom (U 1) w - NormedSpace.exp (Complex.I • laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) * |
| groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)‖ ≤ |
| 3*K*Real.sqrt ((freeCocycle h w - Finsupp.single 1 1).sum (fun _ r => r^2)) := by |
| constructor |
| · intro j |
| have hd := generator_displacement (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) h U hX |
| (fun j => 3*K*Real.sqrt ((h j).sum (fun _ r => r^2))) |
| (fun t j => hbound t (h j)) j |
| rw [hU₀] at hd |
| exact hd |
| · intro w |
| have hd := word_displacement (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) hS h U hX w |
| (3*K*Real.sqrt ((freeCocycle h w - Finsupp.single 1 1).sum (fun _ r => r^2))) |
| (fun t => hbound t (freeCocycle h w - Finsupp.single 1 1)) |
| rwa [hU₀, wordHom_active] at hd |
|
|
| theorem displacement (h : ι → FreeGroup ι →₀ ℝ) (l : ℤ →₀ ℂ) (h0 : l 0 = 0) |
| (hS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l)) |
| (K : ℝ) (hK : 0 ≤ K) |
| (hn : ‖laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l‖ ≤ K) : |
| ∃ β : Ambient ι κ ≃⋆ₐ[ℂ] Ambient ι κ, |
| (∀ T, canonicalTrace (β T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) |
| (groupUltraweakTopology (FreeGroup (Option (ι ⊕ κ)))) β ∧ |
| β (groupUnitary (FreeGroup.of none)) = groupUnitary (FreeGroup.of none) ∧ |
| (∀ k : κ, β (groupUnitary (FreeGroup.of (some (Sum.inr k)))) = |
| groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧ |
| (∀ i : ι, ‖β (groupUnitary (FreeGroup.of (some (Sum.inl i) : Option (ι ⊕ κ)))) - |
| groupUnitary (FreeGroup.of (some (Sum.inl i) : Option (ι ⊕ κ)))‖ ≤ |
| 3*K*Real.sqrt ((h i).sum (fun _ r => r^2))) ∧ |
| ∀ w : FreeGroup ι, ‖β (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) - |
| NormedSpace.exp (Complex.I • laurentValue (generatorUnitary (none : Option (ι ⊕ κ))) l) * |
| groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)‖ ≤ |
| 3*K*Real.sqrt ((freeCocycle h w - Finsupp.single 1 1).sum (fun _ r => r^2)) := by |
| obtain ⟨U,hU₀,hX,hbound,β,hβt,hβc,hβ⟩ := bounded_flow (ι := ι) (κ := κ) h l h0 hS K hK hn |
| obtain ⟨hgen,hword⟩ := displacements_from_bounds (ι := ι) (κ := κ) h l hS K U hU₀ hX hbound |
| refine ⟨β,hβt,hβc,?_,?_,?_,?_⟩ |
| · exact (hβ (FreeGroup.of (none : Option (ι ⊕ κ)))).trans (wordHom_of _ _) |
| · intro k |
| exact (hβ (FreeGroup.of (some (Sum.inr k) : Option (ι ⊕ κ)))).trans (wordHom_of _ _) |
| · intro i |
| rw [hβ] |
| simpa only [wordHom_of,extendUnitary,Option.elim_some,tuple,Sum.elim_inl] using hgen i |
| · intro w |
| rw [hβ] |
| simpa only [wordHom_tuple_active] using hword w |
|
|
| end |
|
|
| open UnitaryEvaluation |
|
|
| theorem one_step_from_laurent {κ : Type uκ} [DecidableEq κ] |
| (n : ℕ) {ε : ℝ} (hε : 0 < ε) |
| (l : ℤ →₀ ℂ) (hl0 : l 0 = 0) |
| (hlS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (Fin n ⊕ κ))) l)) |
| (hln : ‖laurentValue (generatorUnitary (none : Option (Fin n ⊕ κ))) l‖ ≤ 2*(Real.pi+1)) |
| (z : unitary ℂ) (hze : twoNorm (NormedSpace.exp (Complex.I • laurentValue |
| (generatorUnitary (none : Option (Fin n ⊕ κ))) l) - |
| (z : ℂ) • groupUnitary (FreeGroup.of none)) < ε/2) |
| (w : FreeGroup (Fin n)) (h : Coeff (FreeGroup (Fin n))) |
| (hh : coefficientNorm h < (ε/2)/(3*(2*(Real.pi+1)))) |
| (hw : coefficientNorm (freeCocycle (firstCoefficients h) w-Finsupp.single 1 1) < |
| (ε/2)/(3*(2*(Real.pi+1)))) : |
| ∃ α : Ambient (Fin n) κ ≃⋆ₐ[ℂ] Ambient (Fin n) κ, |
| (∀ T, canonicalTrace (α T) = canonicalTrace T) ∧ |
| (∀ k : κ, α (groupUnitary (FreeGroup.of (some (Sum.inr k)))) = |
| groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧ |
| ∃ w : FreeGroup (Fin n), |
| (∀ i : Fin n, ‖α (groupUnitary (FreeGroup.of (some (Sum.inl i)))) - |
| groupUnitary (FreeGroup.of (some (Sum.inl i)))‖ < ε) ∧ |
| twoNorm (α (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) - |
| groupUnitary (FreeGroup.of none)) < ε := by |
| let K : ℝ := 2*(Real.pi+1) |
| have hK : 0 < K := by dsimp [K]; positivity |
| obtain ⟨β,hβt,_hβn,_hβC,hβpass,hβgen,hβword⟩ := |
| displacement (κ := κ) (firstCoefficients h) l hl0 hlS K hK.le hln |
| obtain ⟨γ,hγt,_hγn,_hγi,hγgen,hγC⟩ := |
| gauged_nielsen_iso (ι := Fin n ⊕ κ) (FreeGroup.map Sum.inl w) z |
| have hmap : FreeGroup.map (Option.some : Fin n ⊕ κ → Option (Fin n ⊕ κ)) |
| (FreeGroup.map Sum.inl w) = FreeGroup.map (fun i => some (Sum.inl i)) w := by |
| have he : (FreeGroup.map (Option.some : Fin n ⊕ κ → Option (Fin n ⊕ κ))).comp |
| (FreeGroup.map Sum.inl) = FreeGroup.map (fun i => some (Sum.inl i)) := by |
| apply FreeGroup.ext_hom |
| intro i |
| simp |
| exact DFunLike.congr_fun he w |
| rw [hmap] at hγC |
| refine ⟨β.trans γ.symm,?_,?_,w,?_,?_⟩ |
| · intro T |
| change canonicalTrace (γ.symm (β T)) = canonicalTrace T |
| rw [trace_symm γ hγt,hβt] |
| · intro k |
| change γ.symm (β (groupUnitary (FreeGroup.of (some (Sum.inr k))))) = _ |
| rw [hβpass,← hγgen (Sum.inr k),γ.symm_apply_apply] |
| exact (hγgen (Sum.inr k)).symm |
| · intro i |
| exact one_step_generator_estimate β γ _ (hγgen (Sum.inl i)) hK hε (hβgen i) |
| ((firstCoefficients_norm_le h i).trans_lt hh) |
| · exact one_step_word_estimate β γ hγt |
| (generatorUnitary (none : Option (Fin n ⊕ κ))) |
| (groupUnitaryUHom (FreeGroup (Option (Fin n ⊕ κ))) |
| (FreeGroup.map (fun i => some (Sum.inl i)) w)) |
| _ z hγC hK (hβword w) hw hze |
|
|
| theorem one_step_of_coefficient_estimate {κ : Type uκ} [DecidableEq κ] (n : ℕ) |
| (hc : ∀ η : ℝ, 0 < η → ∃ (w : FreeGroup (Fin n)) (h : Coeff (FreeGroup (Fin n))), |
| coefficientNorm h < η ∧ coefficientNorm (freeCocycle (firstCoefficients h) w-Finsupp.single 1 1) < η) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ α : Ambient (Fin n) κ ≃⋆ₐ[ℂ] Ambient (Fin n) κ, |
| (∀ T, canonicalTrace (α T) = canonicalTrace T) ∧ |
| (∀ k : κ, α (groupUnitary (FreeGroup.of (some (Sum.inr k)))) = |
| groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧ |
| ∃ w : FreeGroup (Fin n), |
| (∀ i : Fin n, ‖α (groupUnitary (FreeGroup.of (some (Sum.inl i)))) - |
| groupUnitary (FreeGroup.of (some (Sum.inl i)))‖ < ε) ∧ |
| twoNorm (α (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) - |
| groupUnitary (FreeGroup.of none)) < ε := by |
| obtain ⟨l,hl0,hlS,hln,z,hze⟩ := haar_centered_laurent |
| (generatorUnitary (none : Option (Fin n ⊕ κ))) (generator_haar _) (half_pos hε) |
| rw [← laurentValue_eq_value] at hlS hln hze |
| obtain ⟨w,h,hh,hw⟩ := hc ((ε/2)/(3*(2*(Real.pi+1)))) (by positivity) |
| exact one_step_from_laurent n hε l hl0 hlS hln z hze w h hh hw |
|
|
| theorem one_step {κ : Type uκ} [DecidableEq κ] (n : ℕ) (hn : 3 ≤ n) |
| {ε : ℝ} (hε : 0 < ε) : |
| ∃ α : Ambient (Fin n) κ ≃⋆ₐ[ℂ] Ambient (Fin n) κ, |
| (∀ T, canonicalTrace (α T) = canonicalTrace T) ∧ |
| (∀ k : κ, α (groupUnitary (FreeGroup.of (some (Sum.inr k)))) = |
| groupUnitary (FreeGroup.of (some (Sum.inr k)))) ∧ |
| ∃ w : FreeGroup (Fin n), |
| (∀ i : Fin n, ‖α (groupUnitary (FreeGroup.of (some (Sum.inl i)))) - |
| groupUnitary (FreeGroup.of (some (Sum.inl i)))‖ < ε) ∧ |
| twoNorm (α (groupUnitary (FreeGroup.map (fun i => some (Sum.inl i)) w)) - |
| groupUnitary (FreeGroup.of none)) < ε := one_step_of_coefficient_estimate n (smallCocycle n hn) hε |
|
|
| end FreeGroupFactorMain.FiniteActive |
|
|
| end |
|
|
| end OAI |
|
|
