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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 | 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
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