| 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') |
|
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| end CStarAbsorption |
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| end |
| end |
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| end OAI |
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