| import Mathlib |
| import OAI.Analysis.FreeWords.ConcreteCornerFactor |
| import OAI.Analysis.FreeWords.ConcreteCornerTrace |
| import OAI.Analysis.FreeWords.LocalCompression |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKMatrixQuantization |
|
|
| open KKTraceTransfer KKFreeLiftFrontier KKProjectionSupport KKConcreteCorner |
|
|
| noncomputable section |
|
|
| theorem corner_small_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) (hx : ∀ i, τ.map (x i)=0) |
| (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))^2 < ε*(τ.map q).re := by |
| have hpCLM : IsStarProjection (p : H →L[ℂ] H) := hp.map M.toStarSubalgebra.subtype |
| let S : Submodule ℂ H := (p.val : H →ₗ[ℂ] H).range |
| let : IsClosed (S : Set H) := ContinuousLinearMap.IsIdempotentElem.isClosed_range hpCLM.isIdempotentElem |
| let : CompleteSpace S := IsClosed.completeSpace_coe |
| have heq : S.starProjection=p.val := by |
| obtain ⟨h,heq⟩ := isStarProjection_iff_eq_starProjection_range.mp hpCLM |
| exact heq.symm |
| have hmem : S.starProjection ∈ M := heq ▸ p.property |
| have hproj : projection S M hmem=p := Subtype.ext heq |
| have hproj0 : projection S M hmem≠0 := hproj ▸ hp0 |
| let C := cornerAlgebra S M |
| let σ := cornerTrace S M hmem τ hproj0 |
| let y : ι → ConcreteAlgebra C := fun i => restrict S M hmem (x i) |
| have hy (i : ι) : liftLinear S M hmem (y i)=x i := by |
| change liftLinear S M hmem (restrict S M hmem (x i))=x i |
| rw [lift_restrict,hproj,hs] |
| have hyτ (i : ι) : σ.map (y i)=0 := by |
| change (cornerTrace S M hmem τ hproj0).map (y i)=0 |
| rw [cornerTrace_apply,hy,hx,mul_zero] |
| let t : ℝ := (τ.map p).re |
| have ht : 0<t := trace_projection_pos τ hp hp0 |
| let δ : ℝ := min ε (ε/t) |
| have hδ : 0<δ := lt_min hε (div_pos hε ht) |
| have hδε : δ≤ε := min_le_left _ _ |
| have hδt : δ≤ε/t := min_le_right _ _ |
| obtain ⟨Q,hQ,hQ0,hQtrace,hQsmall⟩ := local_small_compression C σ |
| (cornerAlgebra_factor S M hmem τ hF hproj0) (cornerAlgebra_diffuse S M hmem hD) |
| y hyτ δ hδ |
| let q := liftLinear S M hmem Q |
| have hq : IsStarProjection q := lift_projection S M hmem Q hQ |
| have hq0 : q≠0 := by |
| intro hh |
| exact hQ0 (lift_injective S M hmem (hh.trans (map_zero _).symm)) |
| have hqp : q*p=q := by |
| rw [← hproj] |
| exact (lift_supported S M hmem Q).2 |
| have htrace : (σ.map Q).re=t⁻¹*(τ.map q).re := by |
| change ((cornerTrace S M hmem τ hproj0).map Q).re=_ |
| rw [cornerTrace_re,hproj] |
| have hnorm (i : ι) : (traceNorm σ (Q*y i*Q))^2=t⁻¹*(traceNorm τ (q*x i*q))^2 := by |
| change (traceNorm (cornerTrace S M hmem τ hproj0) (Q*y i*Q))^2=_ |
| rw [cornerTrace_norm_sq,hproj,lift_mul,lift_mul,hy] |
| rw [htrace] at hQtrace hQsmall |
| simp_rw [hnorm] at hQsmall |
| rw [← Finset.mul_sum] at hQsmall |
| have hqτ : 0 < (τ.map q).re := trace_projection_pos τ hq hq0 |
| refine ⟨q,hq,hq0,hqp,?_,?_⟩ |
| · have h := mul_lt_mul_of_pos_left hQtrace ht |
| have hcan : t*(t⁻¹*(τ.map q).re)=(τ.map q).re := by |
| rw [← mul_assoc,mul_inv_cancel₀ ht.ne',one_mul] |
| rw [hcan] at h |
| have htδ : t*δ≤ε := by |
| have hh := (le_div_iff₀ ht).mp hδt |
| nlinarith |
| exact h.trans_le htδ |
| · have hsmall : ∑ i,(traceNorm τ (q*x i*q))^2 < δ*(τ.map q).re := by |
| have hi : 0 < t⁻¹ := inv_pos.mpr ht |
| have he : δ*(t⁻¹*(τ.map q).re)=t⁻¹*(δ*(τ.map q).re) := by ring |
| rw [he] at hQsmall |
| nlinarith |
| exact hsmall.trans_le (mul_le_mul_of_nonneg_right hδε hqτ.le) |
| |
| end |
| |
| end KKMatrixQuantization |
| end |
| |
| end OAI |
| |
