| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.MatrixAlgebras.MatrixPartialFlips |
| import OAI.Analysis.JiangSu.CStarAlgebras.TwoParameterPartition |
|
|
| namespace OAI |
|
|
|
|
| noncomputable section |
| namespace JiangSuConstruction |
| open MatrixTensor Pureness |
| open scoped CStarAlgebra ComplexOrder TensorProduct |
|
|
| variable {p q r s : ℕ} |
|
|
| lemma dropFiber_zero : dropFiber p q leftEndpoint=(leftAmplification p q).range := by |
| simp [dropFiber] |
| lemma dropFiber_one : dropFiber p q rightEndpoint=(rightAmplification p q).range := by |
| have hn : rightEndpoint≠leftEndpoint := by |
| intro h |
| have := congrArg Subtype.val h |
| norm_num [rightEndpoint,leftEndpoint] at this |
| simp [dropFiber,hn] |
|
|
| def cutoffFiber (t : Interval) (ρ : C(Interval,ℝ)) |
| (h0 : t≠leftEndpoint → ρ leftEndpoint=0) |
| (h1 : t≠rightEndpoint → ρ rightEndpoint=0) : |
| dropFiber p q t →ₗ[ℂ] DimensionDrop p q where |
| toFun a := ⟨⟨fun x => (ρ x:ℂ) • a.val,by fun_prop⟩,by |
| constructor |
| · by_cases ht : t=leftEndpoint |
| · subst t |
| exact (leftAmplification p q).range.smul_mem (by simpa only [dropFiber_zero] using a.property) _ |
| · change (ρ leftEndpoint:ℂ) • a.val∈(leftAmplification p q).range |
| simp [h0 ht] |
| · by_cases ht : t=rightEndpoint |
| · subst t |
| exact (rightAmplification p q).range.smul_mem (by simpa only [dropFiber_one] using a.property) _ |
| · change (ρ rightEndpoint:ℂ) • a.val∈(rightAmplification p q).range |
| simp [h1 ht]⟩ |
| map_add' a b := by |
| apply Subtype.ext; apply ContinuousMap.ext; intro x |
| exact smul_add _ _ _ |
| map_smul' c a := by |
| apply Subtype.ext; apply ContinuousMap.ext; intro x |
| exact smul_comm _ _ _ |
|
|
| local instance (priority := 2000) localDropSemiring (p q : ℕ) : Semiring (DimensionDrop p q) := |
| (inferInstance : CStarAlgebra (DimensionDrop p q)).toSemiring |
|
|
| lemma tensor_cutoff_apply (t u : Interval) (ρ σ : C(Interval,ℝ)) (h0 h1 k0 k1) |
| (z : (dropFiber p q t) ⊗[ℂ] (dropFiber r s u)) (x y : Interval) : |
| ((squareTensorMap p q r s) |
| (TensorProduct.map (cutoffFiber t ρ h0 h1) (cutoffFiber u σ k0 k1) z)).val (x,y)= |
| (ρ x*σ y) • fiberMap (dropFiber p q t) (dropFiber r s u) z := by |
| change (sectionMap (dimensionDrop p q) (dimensionDrop r s) _) (x,y) = _ |
| induction z using TensorProduct.inductionOn with |
| | tmul a b => |
| change left ((ρ x:ℂ) • a.val)*right ((σ y:ℂ) • b.val)= |
| (ρ x*σ y) • (left a.val*right b.val) |
| rw [map_smul,map_smul,smul_mul_assoc,mul_smul_comm,smul_smul,←Complex.ofReal_mul, |
| Complex.coe_smul] |
| rfl |
| | add _ _ hz hw => |
| rw [map_add,map_add,ContinuousMap.add_apply,hz,hw,map_add,smul_add] |
|
|
| lemma convex_sum_error {ι : Type*} [Fintype ι] {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| (w : ι → ℝ) (v : ι → E) (a : E) (hw : ∀ i,0≤w i) (hs : ∑ i,w i=1) |
| {δ : ℝ} (hδ : 0≤δ) (hv : ∀ i,w i≠0 → ‖v i-a‖≤δ) : |
| ‖∑ i,w i • v i-a‖≤δ := by |
| classical |
| have he : ∑ i,w i • v i-a=∑ i,w i • (v i-a) := by |
| simp only [smul_sub,Finset.sum_sub_distrib,←Finset.sum_smul,hs,one_smul] |
| rw [he] |
| calc |
| _ ≤ ∑ i,‖w i • (v i-a)‖ := norm_sum_le _ _ |
| _ ≤ ∑ i,w i*δ := by |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases hi : w i=0 |
| · simp only [hi,zero_smul,norm_zero] |
| exact mul_nonneg (le_refl 0) hδ |
| · rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (hw i)] |
| exact mul_le_mul_of_nonneg_left (hv i hi) (hw i) |
| _ = δ := by rw [←Finset.sum_mul,hs,one_mul] |
|
|
| theorem squareTensorMap_dense : DenseRange (squareTensorMap p q r s) := by |
| classical |
| apply Metric.denseRange_iff.mpr |
| intro f ε hε |
| obtain ⟨S,T,ρ,σ,hρ,hσ,hs,ht,hx0,hx1,hy0,hy1,hf⟩ := |
| EndpointPartition.exists_rectangular_weights leftEndpoint rightEndpoint |
| leftEndpoint rightEndpoint f.val (show 0<ε/2 by positivity) |
| have hsample (i : S) (j : T) : ∃ z : dropFiber p q i.val ⊗[ℂ] dropFiber r s j.val, |
| fiberMap (dropFiber p q i.val) (dropFiber r s j.val) z=f.val (i.val,j.val) := |
| f.property i.val j.val |
| choose z hz using hsample |
| let Z := ∑ i,∑ j,TensorProduct.map (cutoffFiber i.val (ρ i) (hx0 i) (hx1 i)) |
| (cutoffFiber j.val (σ j) (hy0 j) (hy1 j)) (z i j) |
| refine ⟨Z,?_⟩ |
| have heval (x y : Interval) : (squareTensorMap p q r s Z).val (x,y)= |
| ∑ i,∑ j,(ρ i x*σ j y) • f.val (i.val,j.val) := by |
| change ((squareSections p q r s).subtype (squareTensorMap p q r s Z)) (x,y)=_ |
| simp only [Z,map_sum,ContinuousMap.sum_apply] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| change (squareTensorMap p q r s _).val (x,y)=_ |
| rw [tensor_cutoff_apply,hz] |
| have hnorm : ‖squareTensorMap p q r s Z-f‖≤ε/2 := by |
| change ‖(squareTensorMap p q r s Z).val-f.val‖≤ε/2 |
| apply (ContinuousMap.norm_le _ (by positivity)).mpr |
| rintro ⟨x,y⟩ |
| change ‖(squareTensorMap p q r s Z).val (x,y)-f.val (x,y)‖≤ε/2 |
| rw [heval,←Finset.sum_product'] |
| apply convex_sum_error (fun ij : S×T => ρ ij.1 x*σ ij.2 y) |
| (fun ij => f.val (ij.1.val,ij.2.val)) (f.val (x,y)) |
| (fun ij => mul_nonneg (hρ _ _) (hσ _ _)) |
| · rw [Fintype.sum_prod_type] |
| simp only [←Finset.mul_sum,ht,mul_one,hs] |
| · positivity |
| · intro ij hij |
| have hi : ρ ij.1 x≠0 := (mul_ne_zero_iff.mp hij).1 |
| have hj : σ ij.2 y≠0 := (mul_ne_zero_iff.mp hij).2 |
| rw [norm_sub_rev] |
| exact (hf ij.1 ij.2 x y hi hj).le |
| rw [dist_eq_norm,norm_sub_rev] |
| exact hnorm.trans_lt (by linarith) |
| |
| end JiangSuConstruction |
| |
| end |
| |
| |
| noncomputable section |
| namespace JiangSuConstruction |
| open MatrixTensor FourEdgeSurgery |
| open scoped CStarAlgebra ComplexOrder |
| |
| variable {p q : ℕ} |
| |
| lemma leftAmplification_eq : leftAmplification p q=MatrixTensor.left := by rfl |
| lemma rightAmplification_eq : rightAmplification p q=MatrixTensor.right := by rfl |
| |
| def rightUnitary (w : unitary (TensorMat p q)) : unitary (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))) := |
| ⟨MatrixTensor.right (I := Fin p×Fin q) (w:(TensorMat p q)),Unitary.map_mem _ w.property⟩ |
| lemma rightUnitary_continuous : Continuous (rightUnitary (p := p) (q := q)) := |
| continuous_induced_rng.mpr ((map_continuous MatrixTensor.right).comp continuous_subtype_val) |
| @[simp] lemma rightUnitary_one : rightUnitary (p := p) (q := q) 1=1 := by |
| apply Subtype.ext |
| exact map_one _ |
| |
| variable (W : C(Interval×Interval,unitary (TensorMat p q))) (a b : C(Interval,ℝ)) |
| variable (ha0 : a leftEndpoint=1) (ha1 : a rightEndpoint=0) |
| variable (hb0 : b leftEndpoint=0) (hb1 : b rightEndpoint=1) |
| variable (hW0 : W (leftEndpoint,leftEndpoint)=1) |
| variable (hW1 : W (rightEndpoint,rightEndpoint)=1) |
| |
| def surgerySection : C(Interval×Interval,unitary (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q)))) := |
| ⟨fun z => surgery (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) (rightUnitary (W (z.1,leftEndpoint))) |
| (rightUnitary (W (z.1,rightEndpoint))) (rightUnitary (W z)) |
| (a z.1) (b z.1) (a z.2) (b z.2),by |
| apply surgery_continuous |
| · exact pFlow_continuous |
| · exact qFlow_continuous |
| · exact rightUnitary_continuous.comp (W.continuous.comp (continuous_fst.prodMk continuous_const)) |
| · exact rightUnitary_continuous.comp (W.continuous.comp (continuous_fst.prodMk continuous_const)) |
| · exact rightUnitary_continuous.comp W.continuous |
| · exact a.continuous.comp continuous_fst |
| · exact b.continuous.comp continuous_fst |
| · exact a.continuous.comp continuous_snd |
| · exact b.continuous.comp continuous_snd⟩ |
| |
| include ha0 hb0 hW0 in |
| lemma surgerySection_left_zero (t : Interval) : |
| surgerySection W a b (leftEndpoint,t)=(pFlow (I := Fin p) (J := Fin q)) (b t)*rightUnitary (W (leftEndpoint,t))*(pFlow (I := Fin p) (J := Fin q)) 1 := by |
| change surgery (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) _ _ _ _ _ _ _=_ |
| rw [ha0,hb0,hW0,rightUnitary_one] |
| exact surgery_left_zero (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) (by simpa only [qFlow_one] using (partialQ_sq (I := Fin p) (J := Fin q))) _ _ _ _ |
| include ha1 hb1 hW1 in |
| lemma surgerySection_left_one (t : Interval) : |
| surgerySection W a b (rightEndpoint,t)=(qFlow (I := Fin p) (J := Fin q)) (a t)*rightUnitary (W (rightEndpoint,t))*(qFlow (I := Fin p) (J := Fin q)) 1 := by |
| change surgery (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) _ _ _ _ _ _ _=_ |
| rw [ha1,hb1,hW1,rightUnitary_one] |
| exact surgery_left_one (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) (by simpa only [pFlow_one] using (partialP_sq (I := Fin p) (J := Fin q))) |
| flows_commute _ _ _ _ |
| include ha0 hb0 in |
| lemma surgerySection_right_zero (s : Interval) : |
| surgerySection W a b (s,leftEndpoint)= |
| ((qFlow (I := Fin p) (J := Fin q)) 1*rightUnitary (W (s,leftEndpoint))*(qFlow (I := Fin p) (J := Fin q)) 1)*(pFlow (I := Fin p) (J := Fin q)) 1*(pFlow (I := Fin p) (J := Fin q)) (b s) := by |
| change surgery (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) _ _ _ _ _ _ _=_ |
| rw [ha0,hb0] |
| exact surgery_right_zero (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) flows_commute _ _ _ _ |
| include ha1 hb1 in |
| lemma surgerySection_right_one (s : Interval) : |
| surgerySection W a b (s,rightEndpoint)= |
| ((pFlow (I := Fin p) (J := Fin q)) 1*rightUnitary (W (s,rightEndpoint))*(pFlow (I := Fin p) (J := Fin q)) 1)*(qFlow (I := Fin p) (J := Fin q)) 1*(qFlow (I := Fin p) (J := Fin q)) (a s) := by |
| change surgery (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) _ _ _ _ _ _ _=_ |
| rw [ha1,hb1] |
| exact surgery_right_one (pFlow (I := Fin p) (J := Fin q)) (qFlow (I := Fin p) (J := Fin q)) flows_commute _ _ _ _ |
| |
| lemma q_right_mem (w : unitary (TensorMat p q)) : |
| (((qFlow (I := Fin p) (J := Fin q)) 1*rightUnitary w*(qFlow (I := Fin p) (J := Fin q)) 1 : unitary (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q)))) : (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))∈fiberAlgebra (MatrixTensor.right (I := Fin p) (J := Fin q)).range (MatrixTensor.left (I := Fin p) (J := Fin q)).range := by |
| change ((qFlow (I := Fin p) (J := Fin q)) 1:(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))*MatrixTensor.right (w:(TensorMat p q))*((qFlow (I := Fin p) (J := Fin q)) 1:(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))∈_ |
| simp only [qFlow_one] |
| simpa only [unitaryConjugation,StarAlgHom.coe_mk,partialQ_star] using partialQ_conjugate_right_mem (w:(TensorMat p q)) |
| lemma p_right_mem (w : unitary (TensorMat p q)) : |
| (((pFlow (I := Fin p) (J := Fin q)) 1*rightUnitary w*(pFlow (I := Fin p) (J := Fin q)) 1 : unitary (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q)))) : (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))∈fiberAlgebra (MatrixTensor.left (I := Fin p) (J := Fin q)).range (MatrixTensor.right (I := Fin p) (J := Fin q)).range := by |
| change ((pFlow (I := Fin p) (J := Fin q)) 1:(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))*MatrixTensor.right (w:(TensorMat p q))*((pFlow (I := Fin p) (J := Fin q)) 1:(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))∈_ |
| simp only [pFlow_one] |
| simpa only [unitaryConjugation,StarAlgHom.coe_mk,partialP_star] using partialP_conjugate_right_mem (w:(TensorMat p q)) |
| |
| include ha0 ha1 hb0 hb1 hW0 hW1 in |
| |
| lemma surgerySection_mem (s t : Interval) : |
| (surgerySection W a b (s,t):(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))∈fiberAlgebra (dropFiber p q s) (dropFiber p q t) := by |
| have hP (r : ℝ) := pFlow_mem (I := Fin p) (J := Fin q) r |
| have hQ (r : ℝ) := qFlow_mem (I := Fin p) (J := Fin q) r |
| by_cases hs0 : s=leftEndpoint |
| · subst s |
| rw [surgerySection_left_zero W a b ha0 hb0 hW0] |
| rw [dropFiber_zero,leftAmplification_eq] |
| by_cases ht0 : t=leftEndpoint |
| · subst t |
| rw [hb0,hW0,rightUnitary_one,(pFlow (I := Fin p) (J := Fin q)).map_zero,one_mul,one_mul, |
| dropFiber_zero,leftAmplification_eq] |
| exact hP 1 |
| · by_cases ht1 : t=rightEndpoint |
| · subst t |
| rw [hb1,dropFiber_one,rightAmplification_eq] |
| exact p_right_mem _ |
| · rw [dropFiber,ite_eq_right ht0,ite_eq_right ht1] |
| apply mul_mem |
| · apply mul_mem |
| · exact fiberAlgebra_mono le_rfl le_top (hP _) |
| · exact right_mem_fiber (MatrixTensor.left (I := Fin p) (J := Fin q)).range ⊤ (by trivial) |
| · exact fiberAlgebra_mono le_rfl le_top (hP _) |
| · by_cases hs1 : s=rightEndpoint |
| · subst s |
| rw [surgerySection_left_one W a b ha1 hb1 hW1] |
| rw [dropFiber_one,rightAmplification_eq] |
| by_cases ht0 : t=leftEndpoint |
| · subst t |
| rw [ha0,dropFiber_zero,leftAmplification_eq] |
| exact q_right_mem _ |
| · by_cases ht1 : t=rightEndpoint |
| · subst t |
| rw [ha1,hW1,rightUnitary_one,(qFlow (I := Fin p) (J := Fin q)).map_zero,one_mul,one_mul, |
| dropFiber_one,rightAmplification_eq] |
| exact hQ 1 |
| · rw [dropFiber,ite_eq_right ht0,ite_eq_right ht1] |
| apply mul_mem |
| · apply mul_mem |
| · exact fiberAlgebra_mono le_rfl le_top (hQ _) |
| · exact right_mem_fiber (MatrixTensor.right (I := Fin p) (J := Fin q)).range ⊤ (by trivial) |
| · exact fiberAlgebra_mono le_rfl le_top (hQ _) |
| · rw [dropFiber,ite_eq_right hs0,ite_eq_right hs1] |
| by_cases ht0 : t=leftEndpoint |
| · subst t |
| rw [surgerySection_right_zero W a b ha0 hb0, |
| dropFiber_zero,leftAmplification_eq] |
| apply mul_mem |
| · apply mul_mem |
| · exact fiberAlgebra_mono le_top le_rfl (q_right_mem _) |
| · exact fiberAlgebra_mono le_top le_rfl (hP _) |
| · exact fiberAlgebra_mono le_top le_rfl (hP _) |
| · by_cases ht1 : t=rightEndpoint |
| · subst t |
| rw [surgerySection_right_one W a b ha1 hb1, |
| dropFiber_one,rightAmplification_eq] |
| apply mul_mem |
| · apply mul_mem |
| · exact fiberAlgebra_mono le_top le_rfl (p_right_mem _) |
| · exact fiberAlgebra_mono le_top le_rfl (hQ _) |
| · exact fiberAlgebra_mono le_top le_rfl (hQ _) |
| · rw [dropFiber,ite_eq_right ht0,ite_eq_right ht1,fiberAlgebra_top_top] |
| trivial |
| |
| def squareUnitary (U : C(Interval×Interval,unitary (MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))) |
| (hU : ∀ s t,(U (s,t):(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q))))∈fiberAlgebra (dropFiber p q s) (dropFiber p q t)) : |
| unitary (Square p q p q) := |
| ⟨⟨⟨fun z => (U z:(MatrixTensor.Mat (MatrixTensor.Four (Fin p) (Fin q)))),continuous_subtype_val.comp U.continuous⟩,hU⟩,by |
| constructor |
| · apply Subtype.ext |
| apply ContinuousMap.ext |
| intro z |
| exact (U z).property.1 |
| · apply Subtype.ext |
| apply ContinuousMap.ext |
| intro z |
| exact (U z).property.2⟩ |
| |
| def surgeryUnitary : unitary (Square p q p q) := |
| squareUnitary (surgerySection W a b) (surgerySection_mem W a b ha0 ha1 hb0 hb1 hW0 hW1) |
| |
| end JiangSuConstruction |
| |
| end |
| |
| end OAI |
| |
