File size: 5,096 Bytes
67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import Mathlib
import OAI.Analysis.FreeWords.FreePreimage
import OAI.Analysis.FreeWords.VariableUltraBundle
namespace OAI
noncomputable section
namespace KKVariableUltra
open KKTraceTransfer KKFreeLiftFrontier KKUltraTrace
open Filter Topology
open scoped ComplexOrder
noncomputable section
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
variable (R : ℕ → StarSubalgebra ℂ A) [∀ j, IsClosed (R j : Set A)]
local instance coordExtractOrder : PartialOrder (coordinateAlgebra R) := CStarAlgebra.spectralOrder _
local instance coordinateFreeExtractLE : LE (coordinateAlgebra R) := (coordExtractOrder R).toLE
local instance coordinateFreeExtractPreorder : Preorder (coordinateAlgebra R) := (coordExtractOrder R).toPreorder
local instance coordinateFreeExtractSOR : StarOrderedRing (coordinateAlgebra R) := CStarAlgebra.spectralOrderedRing _
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
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
end KKVariableUltra
end
end OAI
|