| import Mathlib |
| import OAI.Analysis.FreeWords.FreePreimage |
| import OAI.Analysis.FreeWords.VariableUltraBundle |
|
|
| namespace OAI |
|
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| noncomputable section |
|
|
| namespace KKVariableUltra |
|
|
| open KKTraceTransfer KKFreeLiftFrontier KKUltraTrace |
|
|
| open Filter Topology |
|
|
| open scoped ComplexOrder |
|
|
| noncomputable section |
|
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| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
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| variable (R : ℕ → StarSubalgebra ℂ A) [∀ j, IsClosed (R j : Set A)] |
|
|
| local instance coordExtractOrder : PartialOrder (coordinateAlgebra R) := CStarAlgebra.spectralOrder _ |
|
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| local instance coordinateFreeExtractLE : LE (coordinateAlgebra R) := (coordExtractOrder R).toLE |
|
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| local instance coordinateFreeExtractPreorder : Preorder (coordinateAlgebra R) := (coordExtractOrder R).toPreorder |
|
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| local instance coordinateFreeExtractSOR : StarOrderedRing (coordinateAlgebra R) := CStarAlgebra.spectralOrderedRing _ |
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|
| def coordinateEval (j : ℕ) : coordinateAlgebra R →⋆ₐ[ℂ] A where |
| toFun x := x.val j |
| map_one' := rfl |
| map_zero' := rfl |
| map_add' _ _ := rfl |
| map_mul' _ _ := rfl |
| commutes' _ := rfl |
| map_star' _ := rfl |
|
|
| omit [PartialOrder A] [StarOrderedRing A] [∀ j, IsClosed (R j : Set A)] in |
| @[simp] theorem coordinateEval_apply (j : ℕ) (x : coordinateAlgebra R) : |
| coordinateEval R j x=x.val j := rfl |
|
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| theorem coordinate_gns_surjective (τ : A →ₚ[ℂ] ℂ) |
| (htr : ∀ x y, τ (x*y)=τ (y*x)) (ω : Ultrafilter ℕ) |
| (hω : (ω : Filter ℕ)≤atTop) : |
| Function.Surjective (gnsBiHom (coordinateTrace R τ ω)) := by |
| intro x |
| have hx := (SetLike.ext_iff.mp (coordinate_gns_range R τ htr ω hω) x.val).mpr x.property |
| obtain ⟨preimage, hpreimage⟩ := hx |
| exact ⟨preimage, Subtype.ext hpreimage⟩ |
|
|
| theorem coordinate_unitary_extract (τ : A →ₚ[ℂ] ℂ) (ω : Ultrafilter ℕ) |
| (hω : (ω : Filter ℕ)≤atTop) |
| (K : Finset ℤ) (F : Finset (List (ℤ × coordinateAlgebra R))) |
| (v : unitary (coordinateAlgebra R)) (ε : ℝ) |
| (hK : ∀ k∈K, k≠0 → ‖coordinateTrace R τ ω ((v^k).val)‖<ε) |
| (hW : ∀ w∈F, ‖coordinateTrace R τ ω ((w.map (fun z => (v^z.1).val*z.2)).prod)‖<ε) |
| (N : ℕ) : |
| ∃ j : ℕ, N≤j ∧ ∃ u : unitary A, u.val∈R j ∧ |
| (∀ k∈K, k≠0 → ‖τ ((u^k).val)‖<ε) ∧ |
| (∀ w∈F, ‖τ ((w.map (fun z => ((u^z.1).val)*(z.2.val j))).prod)‖<ε) := by |
| have hKt : ∀ᶠ j in (ω : Filter ℕ), ∀ k∈K, k≠0 → |
| ‖τ (((Unitary.map (StarMonoidHom.ofClass (coordinateEval R j)) v)^k).val)‖<ε := by |
| apply (Filter.eventually_all_finset K).mpr |
| intro k hk |
| by_cases hk0 : k=0 |
| · exact Eventually.of_forall (fun _ h => (h hk0).elim) |
| · have ht := (tendsto_ultraTraceValue τ ω ((v^k).val.val)).norm |
| have hh := (tendsto_order.mp ht).2 ε (hK k hk hk0) |
| filter_upwards [hh] with j hj |
| intro _ |
| rw [← unitary_map_zpow] |
| change ‖τ ((v^k).val.val j)‖<ε |
| exact hj |
| have hFt : ∀ᶠ j in (ω : Filter ℕ), ∀ w∈F, |
| ‖τ ((w.map (fun z => (((Unitary.map (StarMonoidHom.ofClass (coordinateEval R j)) v)^z.1).val)* |
| (z.2.val j))).prod)‖<ε := by |
| apply (Filter.eventually_all_finset F).mpr |
| intro w hw |
| have ht := (tendsto_ultraTraceValue τ ω ((w.map (fun z => (v^z.1).val*z.2)).prod.val)).norm |
| have hh := (tendsto_order.mp ht).2 ε (hW w hw) |
| filter_upwards [hh] with j hj |
| have hew := unitary_map_word (coordinateEval R j) v w |
| change ‖τ ((coordinateEval R j) ((w.map (fun z => (v^z.1).val*z.2)).prod))‖<ε at hj |
| rw [hew] at hj |
| simpa only [List.map_map,Function.comp_def,coordinateEval_apply] using hj |
| have hN : ∀ᶠ j in (ω : Filter ℕ), N≤j := hω (eventually_ge_atTop N) |
| obtain ⟨j,hjN,hjK,hjF⟩ := (hN.and (hKt.and hFt)).exists |
| exact ⟨j,hjN,Unitary.map (StarMonoidHom.ofClass (coordinateEval R j)) v,v.val.property j,hjK,hjF⟩ |
|
|
| theorem coordinate_free_extract (τ : A →ₚ[ℂ] ℂ) (h1 : τ 1=1) |
| (htr : ∀ x y, τ (x*y)=τ (y*x)) (ω : Ultrafilter ℕ) |
| (hω : (ω : Filter ℕ)≤atTop) |
| (hF : IsFactor (gnsVonNeumann (coordinateTrace R τ ω))) |
| (hD : IsDiffuse (gnsVonNeumann (coordinateTrace R τ ω))) |
| (K : Finset ℤ) (F : Finset (List (ℤ × coordinateAlgebra R))) |
| (hFne : ∀ w∈F, w≠[]) |
| (hcenter : ∀ w∈F, ∀ z∈w, z.1≠0 ∧ coordinateTrace R τ ω z.2=0) |
| (ε : ℝ) (hε : 0<ε) (N : ℕ) : |
| ∃ j : ℕ, N≤j ∧ ∃ u : unitary A, u.val∈R j ∧ |
| (∀ k∈K, k≠0 → ‖τ ((u^k).val)‖<ε) ∧ |
| (∀ w∈F, ‖τ ((w.map (fun z => ((u^z.1).val)*(z.2.val j))).prod)‖<ε) := by |
| obtain ⟨v,hvK,hvF⟩ := finite_free_preimage |
| (gnsVonNeumann (coordinateTrace R τ ω)) |
| (gnsNormalTrace (coordinateTrace R τ ω) (coordinateTrace_one R τ ω h1) |
| (coordinateTrace_tracial R τ ω htr)) hF hD |
| (gnsBiHom (coordinateTrace R τ ω)) (coordinate_gns_surjective R τ htr ω hω) |
| (coordinateTrace R τ ω) (gnsNormalTrace_image _ _ _) K F hFne hcenter ε hε |
| exact coordinate_unitary_extract R τ ω hω K F v ε hvK hvF N |
|
|
| end |
|
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| end KKVariableUltra |
| end |
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| end OAI |
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