| import OAI.Analysis.CharacterCriterion.Definitions |
| import OAI.Analysis.CharacterCriterion.MinimalFunctoriality |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped TensorProduct |
| open scoped InnerProductSpace |
| open scoped BigOperators |
| open Set Module |
| open Topology |
| open NormedSpace |
| open scoped ComplexOrder |
| open scoped ComplexOrder |
| open scoped InnerProductSpace |
| open scoped InnerProductSpace |
| open NormedSpace |
| open scoped TensorProduct InnerProductSpace |
| open UniformSpace |
| variable [KirchbergRordam.GenericRW.Size] |
| local notation "rankNum" => KirchbergRordam.GenericRW.Size.n |
| variable [KirchbergRordam.GenericRW.PauliData] |
| omit [KirchbergRordam.GenericRW.Size] [KirchbergRordam.GenericRW.PauliData] |
| open Filter Topology BoundedContinuousFunction |
| open Filter Topology |
| open Topology |
| open scoped TensorProduct CStarAlgebra |
| open scoped CStarAlgebra ComplexOrder |
| open NormedSpace |
| open scoped ComplexOrder |
| open scoped ComplexOrder |
| open scoped InnerProductSpace |
| open scoped CStarAlgebra ComplexOrder |
| open scoped TensorProduct InnerProductSpace |
| open UniformSpace |
|
|
| namespace KirchbergRordam.PrimeProjectionless |
| open scoped CStarAlgebra Matrix |
| open MatrixDiagonalPath MatrixTensor DropMatrixRepresentations |
| variable {ι κ : Type*} [Fintype ι] [instDecidableEqι : DecidableEq ι] [Fintype κ] [DecidableEq κ] |
| def tr (a : CStarMatrix ι ι ℂ) : ℂ := ∑ i, a i i |
| omit instDecidableEqι in |
| lemma tr_continuous.{contextLevel1} |
| {ι : Type contextLevel1} |
| [Fintype ι] |
| [DecidableEq ι] |
| : Continuous (tr : CStarMatrix ι ι ℂ → ℂ) := by |
| apply continuous_finsetSum |
| intro i hi |
| exact (continuous_apply i).comp (continuous_apply i) |
| lemma tr_nat (a : CStarMatrix ι ι ℂ) (ha : IsIdempotentElem a) : |
| ∃ k : ℕ, k≤Fintype.card ι ∧ tr a=(k:ℂ) := by |
| let e : CStarMatrix ι ι ℂ ≃ₐ[ℂ] ((ι → ℂ) →ₗ[ℂ] (ι → ℂ)) := |
| CStarMatrix.ofMatrixStarAlgEquiv.symm.toAlgEquiv.trans Matrix.toLinAlgEquiv' |
| have h : IsIdempotentElem (e a) := by |
| change e a*e a=e a |
| rw [←map_mul,ha.eq] |
| refine ⟨Module.finrank ℂ (LinearMap.range (e a)), ?_, ?_⟩ |
| · simpa using (LinearMap.range (e a)).finrank_le |
| · have ht := (LinearMap.IsIdempotentElem.isProj_range (e a) h).trace |
| change LinearMap.trace ℂ (ι → ℂ) (Matrix.toLin' (CStarMatrix.ofMatrixStarAlgEquiv.symm a)) = _ at ht |
| rw [Matrix.trace_toLin'_eq] at ht |
| exact ht |
| lemma tr_zero (a : CStarMatrix ι ι ℂ) (ha : IsIdempotentElem a) (ht : tr a=0) : a=0 := by |
| let e : CStarMatrix ι ι ℂ ≃ₐ[ℂ] ((ι → ℂ) →ₗ[ℂ] (ι → ℂ)) := |
| CStarMatrix.ofMatrixStarAlgEquiv.symm.toAlgEquiv.trans Matrix.toLinAlgEquiv' |
| have h : IsIdempotentElem (e a) := by |
| change e a*e a=e a |
| rw [←map_mul,ha.eq] |
| apply e.injective |
| rw [map_zero] |
| apply LinearMap.IsIdempotentElem.eq_zero_of_trace_eq_zero h |
| change LinearMap.trace ℂ (ι → ℂ) (Matrix.toLin' (CStarMatrix.ofMatrixStarAlgEquiv.symm a)) = 0 |
| rw [Matrix.trace_toLin'_eq] |
| exact ht |
| lemma tr_left (a : CStarMatrix ι ι ℂ) : |
| tr (MatrixTensor.leftMatrix (κ:=κ) a)=(Fintype.card κ:ℂ)*tr a := by |
| simp only [tr,Fintype.sum_prod_type,MatrixBlockReindex.left_apply,ite_true] |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,Finset.mul_sum,mul_one] |
| lemma tr_right (a : CStarMatrix κ κ ℂ) : |
| tr (MatrixTensor.rightMatrix (ι:=ι) a)=(Fintype.card ι:ℂ)*tr a := by |
| simp only [tr,Fintype.sum_prod_type,MatrixBlockReindex.right_apply,ite_true] |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,one_mul] |
| lemma tr_constant {X : Type*} [TopologicalSpace X] [PreconnectedSpace X] |
| (f : X → CStarMatrix ι ι ℂ) (hf : Continuous f) (hp : ∀ x, IsIdempotentElem (f x)) |
| (x y : X) : tr (f x)=tr (f y) := by |
| let g : X → ℂ := fun x => tr (f x) |
| have hc : Continuous g := tr_continuous.comp hf |
| have hfin : (Set.range g).Finite := by |
| apply ((Finset.range (Fintype.card ι+1)).finite_toSet.image (fun n : ℕ => (n:ℂ))).subset |
| rintro _ ⟨x,rfl⟩ |
| obtain ⟨k,hk,he⟩ := tr_nat (f x) (hp x) |
| exact ⟨k,Finset.mem_range.mpr (by omega),he.symm⟩ |
| by_contra hn |
| exact hfin.not_infinite ((isPreconnected_range hc).infinite_of_nontrivial |
| ⟨g x,⟨x,rfl⟩,g y,⟨y,rfl⟩,hn⟩) |
| lemma tr_one_sub (a : CStarMatrix ι ι ℂ) : tr (1-a)=(Fintype.card ι:ℂ)-tr a := by |
| simp only [tr] |
| simp |
| lemma prime_idempotent (p q : ℕ) [NeZero p] [NeZero q] (hpq : p.Coprime q) |
| (e : PrimeDimensionDrop.Algebra p q) (he : IsIdempotentElem e) : e=0 ∨ e=1 := by |
| have heval (t : unitInterval) : IsIdempotentElem (e.val t) := by |
| have h := congrArg (fun f : PrimeDimensionDrop.Algebra p q => f.val t) he.eq |
| exact h |
| have hl : IsIdempotentElem (leftEvaluation p q e) := by |
| change leftEvaluation p q e * leftEvaluation p q e = _ |
| rw [←map_mul,he.eq] |
| have hr : IsIdempotentElem (rightEvaluation p q e) := by |
| change rightEvaluation p q e * rightEvaluation p q e = _ |
| rw [←map_mul,he.eq] |
| obtain ⟨k,hk,hke⟩ := tr_nat _ hl |
| obtain ⟨l,hl,hle⟩ := tr_nat _ hr |
| simp only [Fintype.card_fin] at hk hl |
| have ht (t : unitInterval) : tr (e.val t) = tr (e.val 0) := |
| tr_constant e.val e.val.continuous heval t 0 |
| have hzero : tr (e.val 0)=(q:ℂ)*(k:ℂ) := by |
| rw [←leftValue_spec p q e,tr_left,Fintype.card_fin] |
| change (q:ℂ)*tr (leftEvaluation p q e) = _ |
| rw [hke] |
| have hone : tr (e.val 1)=(p:ℂ)*(l:ℂ) := by |
| rw [←rightValue_spec p q e,tr_right,Fintype.card_fin] |
| change (p:ℂ)*tr (rightEvaluation p q e) = _ |
| rw [hle] |
| have hkl : q*k=p*l := by |
| have h := ht 1 |
| rw [hzero,hone] at h |
| exact_mod_cast h.symm |
| have hpk : p ∣ k := hpq.dvd_of_dvd_mul_left (hkl.symm ▸ dvd_mul_right p l) |
| have hkeq : k=0 ∨ k=p := by |
| by_cases h : k=0 |
| · exact Or.inl h |
| · exact Or.inr (le_antisymm hk (Nat.le_of_dvd (Nat.pos_of_ne_zero h) hpk)) |
| rcases hkeq with rfl|rfl |
| · left |
| apply Subtype.ext |
| apply ContinuousMap.ext |
| intro t |
| exact tr_zero (e.val t) (heval t) (by rw [ht t,hzero]; simp) |
| · right |
| have hz : 1-e=0 := by |
| apply Subtype.ext |
| apply ContinuousMap.ext |
| intro t |
| apply tr_zero (1-e.val t) (heval t).one_sub |
| rw [tr_one_sub,ht t,hzero] |
| simp only [Fintype.card_prod,Fintype.card_fin,Nat.cast_mul] |
| ring |
| exact (sub_eq_zero.mp hz).symm |
| |
| end KirchbergRordam.PrimeProjectionless |
| namespace KirchbergRordam.ProjectionRounding |
| open scoped CStarAlgebra |
| |
| def cut (t : ℝ) : ℝ := max 0 (min 1 (2*t-1/2)) |
| lemma cut_continuous : Continuous cut := by unfold cut; fun_prop |
| lemma cut_low {t : ℝ} (h : t≤1/4) : cut t=0 := by |
| unfold cut |
| rw [max_eq_left] |
| exact (min_le_right _ _).trans (by linarith) |
| lemma cut_high {t : ℝ} (h : 3/4≤t) : cut t=1 := by |
| unfold cut |
| rw [min_eq_left (by linarith),max_eq_right (by norm_num)] |
| lemma scalar_round {t : ℝ} (h : |t*t-t|<3/16) : |
| (cut t=0 ∨ cut t=1) ∧ |t-cut t|≤1/4 := by |
| have h' := abs_lt.mp h |
| have hlo : -(1/4:ℝ)<t := by |
| by_contra hn |
| have := sq_nonneg (t+1/4) |
| nlinarith |
| have hhi : t<(5/4:ℝ) := by |
| by_contra hn |
| have := sq_nonneg (t-5/4) |
| nlinarith |
| have hg : t<1/4 ∨ 3/4<t := by |
| by_contra hn |
| push Not at hn |
| have := mul_nonneg (show 0≤t-1/4 by linarith) (show 0≤3/4-t by linarith) |
| nlinarith |
| rcases hg with h|h |
| · rw [cut_low h.le] |
| exact ⟨Or.inl rfl,abs_le.mpr ⟨by linarith,by linarith⟩⟩ |
| · rw [cut_high h.le] |
| exact ⟨Or.inr rfl,abs_le.mpr ⟨by linarith,by linarith⟩⟩ |
| variable {A : Type*} [CStarAlgebra A] |
| lemma polynomial_cfc (a : A) (ha : IsSelfAdjoint a) : |
| cfc (fun t : ℝ => t*t-t) a=a*a-a := by |
| rw [cfc_sub (fun t : ℝ => t*t) (fun t => t) a (by fun_prop) (by fun_prop), |
| cfc_mul (fun t : ℝ => t) (fun t => t) a (by fun_prop) (by fun_prop),cfc_id' ℝ a ha] |
| lemma spectrum_small_defect {a : A} (ha : IsSelfAdjoint a) (h : ‖a*a-a‖<3/16) |
| {t : ℝ} (ht : t∈spectrum ℝ a) : |t*t-t|<3/16 := by |
| have hn := norm_apply_le_norm_cfc (fun t : ℝ => t*t-t) a ht (by fun_prop) ha |
| rw [polynomial_cfc a ha,Real.norm_eq_abs] at hn |
| exact hn.trans_lt h |
| lemma cut_idempotent {a : A} (ha : IsSelfAdjoint a) (h : ‖a*a-a‖<3/16) : |
| IsIdempotentElem (cfc cut a) := by |
| change cfc cut a*cfc cut a=cfc cut a |
| rw [←cfc_mul cut cut a cut_continuous.continuousOn cut_continuous.continuousOn] |
| apply cfc_congr |
| intro t ht |
| rcases (scalar_round (spectrum_small_defect ha h ht)).1 with h|h <;> simp only [h,mul_zero,mul_one] |
| lemma cut_close {a : A} (ha : IsSelfAdjoint a) (h : ‖a*a-a‖<3/16) : |
| ‖a-cfc cut a‖≤1/4 := by |
| calc |
| ‖a-cfc cut a‖ = ‖cfc (fun t : ℝ => t-cut t) a‖ := by |
| rw [cfc_sub (fun t : ℝ => t) cut a (by fun_prop) cut_continuous.continuousOn,cfc_id' ℝ a ha] |
| _ ≤ 1/4 := norm_cfc_le (by norm_num) (fun t ht => |
| (scalar_round (spectrum_small_defect ha h ht)).2) |
| end KirchbergRordam.ProjectionRounding |
| namespace KirchbergRordam.ProjectionRounding |
| open scoped CStarAlgebra |
| variable {A B : Type*} [CStarAlgebra A] [CStarAlgebra B] |
| def part (a : A) : A := (1/2:ℂ) • (a+star a) |
| lemma part_selfAdjoint (a : A) : IsSelfAdjoint (part a) := by |
| change star (part a)=part a |
| simp only [part,star_smul,star_add,star_star] |
| simp [add_comm] |
| lemma part_sub (a b : A) : part (a-b)=part a-part b := by |
| simp only [part,star_sub,sub_add_sub_comm,smul_sub] |
| lemma part_fixed {a : A} (ha : IsSelfAdjoint a) : part a=a := by |
| simp only [part,ha.star_eq,←two_smul ℂ a,smul_smul] |
| norm_num |
| lemma part_map (φ : A →⋆ₐ[ℂ] B) (a : A) : φ (part a)=part (φ a) := by |
| simp only [part,map_smul,map_add,map_star] |
| lemma norm_part_le (a : A) : ‖part a‖≤‖a‖ := by |
| calc |
| ‖part a‖ = (1/2:ℝ)*‖a+star a‖ := by rw [part,norm_smul]; simp |
| _ ≤ (1/2:ℝ)*(‖a‖+‖star a‖) := mul_le_mul_of_nonneg_left (norm_add_le _ _) (by norm_num) |
| _ = ‖a‖ := by rw [norm_star]; ring |
| lemma norm_part_sub_le (a p : A) (hp : IsSelfAdjoint p) : ‖part a-p‖≤‖a-p‖ := by |
| calc |
| ‖part a-p‖=‖part (a-p)‖ := by rw [part_sub,part_fixed hp] |
| _ ≤ _ := norm_part_le _ |
| lemma idempotent_eq_zero_of_norm_lt_one {p : A} (hp : IsIdempotentElem p) (h : ‖p‖<1) : p=0 := by |
| apply norm_eq_zero.mp |
| have hn := norm_mul_le p p |
| rw [hp.eq] at hn |
| have hn0 := norm_nonneg p |
| nlinarith |
| lemma small_defect_of_near (a p : A) (hp : IsStarProjection p) (h : ‖a-p‖<1/100) : |
| ‖a*a-a‖<3/16 := by |
| have hp1 := hp.norm_le p |
| have ha2 : ‖a‖<2 := by |
| have hn : ‖a‖≤‖a-p‖+‖p‖ := by |
| simpa only [sub_add_cancel] using norm_add_le (a-p) p |
| linarith |
| have he : a*a-a=(a-p)*a+p*(a-p)-(a-p) := by |
| have hpp := hp.isIdempotentElem.eq |
| noncomm_ring [hpp] |
| have hn : ‖a*a-a‖≤‖a-p‖*‖a‖+‖p‖*‖a-p‖+‖a-p‖ := by |
| rw [he] |
| exact (norm_sub_le _ _).trans (add_le_add ((norm_add_le _ _).trans |
| (add_le_add (norm_mul_le _ _) (norm_mul_le _ _))) le_rfl) |
| have ha0 := norm_nonneg (a-p) |
| have hn' := mul_le_mul_of_nonneg_left ha2.le ha0 |
| have hn'' := mul_le_mul_of_nonneg_right hp1 ha0 |
| nlinarith |
| lemma projection_trivial_from_approximation (φ : A →⋆ₐ[ℂ] B) (hφ : Isometry φ) |
| (hA : ∀ a : A, IsIdempotentElem a → a=0 ∨ a=1) |
| (a : A) (p : B) (hp : IsStarProjection p) (h : ‖φ a-p‖<1/100) : p=0 ∨ p=1 := by |
| let b := part a |
| have hb : IsSelfAdjoint b := part_selfAdjoint a |
| have hclose : ‖φ b-p‖<1/100 := by |
| rw [part_map] |
| exact (norm_part_sub_le _ _ hp.isSelfAdjoint).trans_lt h |
| have hnorm (x : A) : ‖φ x‖=‖x‖ := hφ.norm_map_of_map_zero (map_zero φ) x |
| have hdefect : ‖b*b-b‖<3/16 := by |
| rw [←hnorm (b*b-b),map_sub,map_mul] |
| exact small_defect_of_near (φ b) p hp hclose |
| have hq := cut_idempotent hb hdefect |
| have hdist : ‖p-φ (cfc cut b)‖<1 := by |
| calc |
| ‖p-φ (cfc cut b)‖ ≤ ‖p-φ b‖+‖φ b-φ (cfc cut b)‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _ |
| _ = ‖φ b-p‖+‖b-cfc cut b‖ := by rw [norm_sub_rev p,←map_sub,hnorm] |
| _ < 1 := by have := cut_close hb hdefect; linarith |
| rcases hA _ hq with hz|ho |
| · left |
| rw [hz,map_zero,sub_zero] at hdist |
| exact idempotent_eq_zero_of_norm_lt_one hp.isIdempotentElem hdist |
| · right |
| rw [ho,map_one,norm_sub_rev] at hdist |
| exact (sub_eq_zero.mp (idempotent_eq_zero_of_norm_lt_one hp.isIdempotentElem.one_sub hdist)).symm |
| end KirchbergRordam.ProjectionRounding |
| namespace KirchbergRordam.BalancedDropLimit |
| open scoped CStarAlgebra |
| |
| |
| theorem projectionless (p : Algebra) (hp : IsStarProjection p) : p=0 ∨ p=1 := by |
| obtain ⟨v,hv⟩ := denseRange_inclusion.exists_dist_lt p (show (0:ℝ)<1/100 by norm_num) |
| have h : ‖inclusion v.1 v.2-p‖<1/100 := by |
| simpa only [dist_eq_norm,norm_sub_rev] using hv |
| exact ProjectionRounding.projection_trivial_from_approximation |
| (inclusion v.1) (inclusion_isometry v.1) |
| (fun e he => PrimeProjectionless.prime_idempotent |
| (BalancedDropMultiplicity.stage v.1) (BalancedDropMultiplicity.stage v.1+1) |
| (BalancedDropMultiplicity.stage_coprime v.1) e he) v.2 p hp h |
| end KirchbergRordam.BalancedDropLimit |
| namespace KirchbergRordam |
| |
| |
| end KirchbergRordam |
| end |
| |
| end OAI |
| |
