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67aee1d | 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 73 74 75 76 | import Mathlib
import OAI.Analysis.FreeWords.BoundedStrongContinuity
import OAI.Analysis.FreeWords.CenteredMatrixDensityDefinitions
import OAI.Analysis.FreeWords.TraceStrongContinuity
namespace OAI
noncomputable section
namespace KKTraceTransfer
open Filter Topology
noncomputable section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
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'
end
end KKTraceTransfer
end
end OAI
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