| import Mathlib |
| import OAI.Analysis.FreeWords.BoundedStrongContinuity |
| import OAI.Analysis.FreeWords.CenteredMatrixDensityDefinitions |
| import OAI.Analysis.FreeWords.TraceStrongContinuity |
|
|
| namespace OAI |
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|
| noncomputable section |
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| namespace KKTraceTransfer |
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|
| open Filter Topology |
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| noncomputable section |
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| theorem centered_matrix_countable_dense [TopologicalSpace.SeparableSpace H] |
| (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) (n : ℕ) : |
| ∃ a : ℕ → Matrix (Fin n) (Fin n) (ConcreteAlgebra M), |
| (∀ j, amplifiedTrace M τ n (a j) = 0) ∧ |
| (∀ j i k, ‖a j i k‖ ≤ 1) ∧ |
| ∀ x : Matrix (Fin n) (Fin n) (ConcreteAlgebra M), |
| amplifiedTrace M τ n x = 0 → (∀ i k, ‖x i k‖ ≤ 1) → |
| ∀ ε : ℝ, 0 < ε → ∃ j, ∀ i k, traceNorm τ (a j i k - x i k) < ε := by |
| classical |
| obtain ⟨D,hD⟩ := TopologicalSpace.exists_dense_seq H |
| let B := {x : Matrix (Fin n) (Fin n) (ConcreteAlgebra M) |
| amplifiedTrace M τ n x = 0 ∧ ∀ i k, ‖x i k‖ ≤ 1} |
| let : Inhabited B := ⟨⟨0,by simp [amplifiedTrace]⟩⟩ |
| let E : B → ((Fin n × Fin n) × ℕ → H) := |
| fun a p => (a.val p.1.1 p.1.2 : H →L[ℂ] H) (D p.2) |
| let : TopologicalSpace B := TopologicalSpace.induced E inferInstance |
| have hE : Topology.IsInducing E := ⟨rfl⟩ |
| let : SecondCountableTopology B := hE.secondCountableTopology |
| have hec : Continuous E := continuous_induced_dom |
| have hc (i k : Fin n) (ξ : H) : |
| Continuous (fun a : B => (a.val i k : H →L[ℂ] H) ξ) := by |
| apply bounded_evals_continuous (fun a : B => (a.val i k : H →L[ℂ] H)) |
| (fun a => a.property.2 i k) D hD |
| intro j |
| exact (continuous_apply ((i,k),j)).comp hec |
| obtain ⟨a,ha⟩ := TopologicalSpace.exists_dense_seq B |
| refine ⟨fun j => (a j).val,fun j => (a j).property.1,fun j => (a j).property.2,?_⟩ |
| intro x hx hb ε hε |
| have ht (i k : Fin n) : |
| Continuous (fun b : B => traceNorm τ (b.val i k - x i k)) := |
| traceNorm_strong_continuous M τ (fun b : B => b.val i k) |
| (fun b => b.property.2 i k) (hc i k) (x i k) (hb i k) |
| let O := {b : B | ∀ i k, traceNorm τ (b.val i k - x i k) < ε} |
| have ho : IsOpen O := by |
| dsimp only [O] |
| simp only [Set.ofPred_forall] |
| apply isOpen_iInter_of_finite |
| intro i |
| apply isOpen_iInter_of_finite |
| intro k |
| exact isOpen_Iio.preimage (ht i k) |
| have hn : O.Nonempty := by |
| refine ⟨⟨x,hx,hb⟩,?_⟩ |
| intro i k |
| change traceNorm τ (x i k-x i k) < ε |
| simpa [traceNorm] using hε |
| obtain ⟨b,hb',j,hj⟩ := ha.inter_open_nonempty O ho hn |
| refine ⟨j,?_⟩ |
| change (a j) ∈ O |
| rw [hj] |
| exact hb' |
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| end |
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| end KKTraceTransfer |
| end |
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| end OAI |
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