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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 | import Mathlib
import OAI.Analysis.FreeWords.CornerCompression
namespace OAI
noncomputable section
namespace KKMatrixQuantization
open KKTraceTransfer KKFreeLiftFrontier KKProjectionSupport
noncomputable section
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
end
end KKMatrixQuantization
end
end OAI
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