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f1f7953 | 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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.CentralSequences.NormUltrapowerMaps
namespace OAI
section
noncomputable section
namespace CStarAbsorption
open Pureness NormUltra NormUltra.QuotientAlgebra
open scoped CStarAlgebra ENNReal TensorProduct
variable {A D B : Type*} [CStarAlgebra A] [CStarAlgebra D] [CStarAlgebra B]
variable (U : Ultrafilter ℕ) (ι : A →⋆ₐ[ℂ] B) (κ : D →⋆ₐ[ℂ] B)
(hic : ∀ a d,Commute (ι a) (κ d))
(φ : D →⋆ₐ[ℂ] QuotientAlgebra (fun _ : ℕ => A) U)
(hφ : ∀ a d,Commute (diagonalHom U a) (φ d))
def preimageTensor : A ⊗[ℂ] D →⋆ₐ[ℂ] QuotientAlgebra (fun _ : ℕ => A) U :=
commutingStarTensorHom (diagonalHom U) φ hφ
theorem finite_tensor_control (z : A ⊗[ℂ] D) :
∃ F : Finset D,∃ C : ℝ,0≤C ∧
∀ u : unitary (QuotientAlgebra (fun _ : ℕ => B) U),
(∀ a,Commute (u : QuotientAlgebra (fun _ : ℕ => B) U) (diagonalHom U (ι a))) →
∀ ε : ℝ,0≤ε →
(∀ d∈F,‖(u : QuotientAlgebra (fun _ : ℕ => B) U)*QuotientAlgebra.map U ι (φ d)*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (κ d)‖≤ε) →
‖(u : QuotientAlgebra (fun _ : ℕ => B) U)*QuotientAlgebra.map U ι (preimageTensor U φ hφ z)*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (commutingStarTensorHom ι κ hic z)‖≤C*ε := by
classical
induction z using TensorProduct.inductionOn with
| tmul a d =>
refine ⟨{d},‖a‖,norm_nonneg _,?_⟩
intro u hu ε hε hf
have hd := hf d (Finset.mem_singleton_self _)
have hcomm := (hu a).eq
have hb : ‖diagonalHom U (ι a)‖≤‖a‖ :=
(NonUnitalStarAlgHom.norm_apply_le (diagonalHom U) (ι a)).trans
(NonUnitalStarAlgHom.norm_apply_le ι a)
change ‖(u : QuotientAlgebra (fun _ : ℕ => B) U)*QuotientAlgebra.map U ι
(diagonalHom U a*φ d)*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (ι a*κ d)‖≤‖a‖*ε
rw [map_mul,map_diagonalHom,map_mul]
have he : (u : QuotientAlgebra (fun _ : ℕ => B) U)*(diagonalHom U (ι a)*QuotientAlgebra.map U ι (φ d))*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (ι a)*diagonalHom U (κ d)=
diagonalHom U (ι a)*((u : QuotientAlgebra (fun _ : ℕ => B) U)*QuotientAlgebra.map U ι (φ d)*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (κ d)) := by
rw [mul_sub,←mul_assoc (u : QuotientAlgebra (fun _ : ℕ => B) U) (diagonalHom U (ι a)),hcomm]
simp only [mul_assoc]
rw [he]
exact (norm_mul_le _ _).trans
(mul_le_mul hb hd (norm_nonneg _) (norm_nonneg _))
| add z w hz hw =>
obtain ⟨F,C,hC,hz⟩ := hz
obtain ⟨G,K,hK,hw⟩ := hw
refine ⟨F∪G,C+K,add_nonneg hC hK,?_⟩
intro u hu ε hε hf
have hz' := hz u hu ε hε (fun d hd => hf d (Finset.mem_union_left _ hd))
have hw' := hw u hu ε hε (fun d hd => hf d (Finset.mem_union_right _ hd))
simp only [map_add]
have he : (u : QuotientAlgebra (fun _ : ℕ => B) U)*
(QuotientAlgebra.map U ι (preimageTensor U φ hφ z)+QuotientAlgebra.map U ι (preimageTensor U φ hφ w))*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-
(diagonalHom U (commutingStarTensorHom ι κ hic z)+diagonalHom U (commutingStarTensorHom ι κ hic w))=
((u : QuotientAlgebra (fun _ : ℕ => B) U)*QuotientAlgebra.map U ι (preimageTensor U φ hφ z)*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (commutingStarTensorHom ι κ hic z))+
((u : QuotientAlgebra (fun _ : ℕ => B) U)*QuotientAlgebra.map U ι (preimageTensor U φ hφ w)*star (u : QuotientAlgebra (fun _ : ℕ => B) U)-diagonalHom U (commutingStarTensorHom ι κ hic w)) := by noncomm_ring
rw [he,add_mul]
exact (norm_add_le _ _).trans (add_le_add hz' hw')
end CStarAbsorption
end
end
end OAI
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