| import Mathlib |
| import OAI.Analysis.FreeWords.CornerCompression |
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| namespace OAI |
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| noncomputable section |
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| namespace KKMatrixQuantization |
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| open KKTraceTransfer KKFreeLiftFrontier KKProjectionSupport |
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| noncomputable section |
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| theorem corner_centered_compression |
| {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| {ι : Type*} [Fintype ι] |
| (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) |
| (hF : IsFactor M) (hD : IsDiffuse M) |
| (p : ConcreteAlgebra M) (hp : IsStarProjection p) (hp0 : p≠0) |
| (x : ι → ConcreteAlgebra M) (hs : ∀ i, p*x i*p=x i) |
| (ε : ℝ) (hε : 0<ε) : |
| ∃ q : ConcreteAlgebra M, IsStarProjection q ∧ q≠0 ∧ q*p=q ∧ (τ.map q).re<ε ∧ |
| ∑ i, (traceNorm τ (q*x i*q-((τ.map p)⁻¹*τ.map (x i)) • q))^2 < ε*(τ.map q).re := by |
| let c (i:ι) : ℂ := (τ.map p)⁻¹*τ.map (x i) |
| let y (i:ι) : ConcreteAlgebra M := x i-c i • p |
| have hy (i:ι) : p*y i*p=y i := by |
| dsimp only [y] |
| rw [mul_sub,sub_mul,mul_smul_comm,smul_mul_assoc,hp.isIdempotentElem.eq, |
| hp.isIdempotentElem.eq,hs] |
| have hτp : τ.map p ≠ 0 := trace_projection_ne_zero τ hp hp0 |
| have hyτ (i:ι) : τ.map (y i)=0 := by |
| change τ.map (x i-c i • p)=0 |
| rw [map_sub,map_smul,smul_eq_mul,sub_eq_zero] |
| dsimp only [c] |
| field_simp |
| obtain ⟨q,hq,hq0,hqp,hqt,hqs⟩ := corner_small_compression M τ hF hD p hp hp0 y hy hyτ ε hε |
| refine ⟨q,hq,hq0,hqp,hqt,?_⟩ |
| have he (i:ι) : q*y i*q=q*x i*q-((τ.map p)⁻¹*τ.map (x i)) • q := by |
| dsimp only [y,c] |
| rw [mul_sub,sub_mul,mul_smul_comm,smul_mul_assoc,hqp,hq.isIdempotentElem.eq] |
| simpa only [he] using hqs |
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| end |
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| end KKMatrixQuantization |
| end |
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| end OAI |
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