| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.MatrixAlgebras.MatrixProjectionTests |
| import OAI.Analysis.JiangSu.MatrixAlgebras.TracialMatrixCorners |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace DirectComparison |
| open Pureness CurrentMain DirectPacking Filter |
| open scoped ComplexOrder |
|
|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable [Nontrivial A] [Nonempty (TracialState A)] (U : Ultrafilter ℕ) |
| local notation "T" => (fun τ : TracialState A => τ) |
| local notation "D" => FamilyUltrapower T U |
| local notation "L" => familyLimitTrace T U |
| local notation "ι" => familyUltrapowerMap T U |
|
|
| theorem comparison_finite_matrix_models (hfree : (U : Filter ℕ)≤atTop) |
| (hsimple : TopologicallySimple A) (hinf : ¬FiniteDimensional ℂ A) |
| (hcmp : StrictComparison A) (hnuc : IsNuclear A) |
| (m : ℕ) (hm : 0 < m) (K : Finset (unitary A)) {ε : ℝ} (hε : 0<ε) : |
| Nonempty (TracialMatrixModel L ι m K ε) := by |
| let := limitTraceFamily (A := A) U |
| let := limitTraceFamily_nonempty (A := A) U |
| obtain ⟨hs,hc,hi⟩ := source_tools U hfree hsimple hinf hcmp |
| obtain ⟨M⟩ := matrix_models_from_packing hnuc ι hs hc hi hm |
| (fun e he => tracial_matrix_corner U hfree hsimple hinf hcmp hnuc m hm he) K hε |
| have h2 (a : D) : familyTwoNorm L a≤familyTwoNorm (fun τ : RestrictedTracialState D => (τ : TracialState D)) a := by |
| apply ciSup_le |
| intro s |
| exact traceTwoNorm_le_family (fun τ : RestrictedTracialState D => (τ : TracialState D)) ⟨L s,⟨s,rfl⟩⟩ a |
| exact ⟨{ |
| entry := M.entry |
| norm_entry := M.norm_entry |
| star_error := fun i j => (h2 _).trans_lt (M.star_error i j) |
| mul_error := fun i j k l => (h2 _).trans_lt (M.mul_error i j k l) |
| unit_error := (h2 _).trans_lt M.unit_error |
| comm_error := fun i j u hu => (h2 _).trans_lt (M.comm_error i j u hu)}⟩ |
|
|
| end DirectComparison |
|
|
| end |
| end |
|
|
| end OAI |
|
|
