| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.Traces.SmallTracePeakIsometry | |
| import OAI.Analysis.JiangSu.Excision.ExcisionUpperCut | |
| import OAI.Analysis.JiangSu.CuntzSemigroup.CuntzControlledSupport | |
| namespace OAI | |
| noncomputable section | |
| namespace CStarExcision | |
| open scoped BigOperators | |
| variable {A : Type*} [NonUnitalCStarAlgebra A] | |
| variable {I : Type*} [Fintype I] | |
| def assembled (d : I → A) (r : A) (c : I → A) : A := ∑ i,d i*r*c i | |
| def coefficientValue (ω : A → ℂ) (d c : I → A) (a : A) : A := | |
| ∑ i,∑ j,ω (star (d i)*a*d j) • (star (c i)*c j) | |
| lemma assembled_square (d c : I → A) (r a : A) : | |
| star (assembled d r c)*a*assembled d r c= | |
| ∑ i,∑ j,star (c i)*(star r*(star (d i)*a*d j)*r)*c j := by | |
| simp only [assembled,star_sum,star_mul,Finset.sum_mul,Finset.mul_sum] | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| simp only [mul_assoc] | |
| lemma coefficient_mul (ω : A → ℂ) (d c : I → A) (a e : A) | |
| (he : ∀ i,e*c i=c i*e) : | |
| coefficientValue ω d c a*e= | |
| ∑ i,∑ j,star (c i)*(ω (star (d i)*a*d j) • e)*c j := by | |
| simp only [coefficientValue,Finset.sum_mul,smul_mul_assoc,mul_smul_comm] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| congr 1 | |
| calc | |
| star (c i)*c j*e=star (c i)*(c j*e) := mul_assoc _ _ _ | |
| _=star (c i)*(e*c j) := by rw [he j] | |
| _=star (c i)*e*c j := (mul_assoc _ _ _).symm | |
| lemma assembly_error_identity (ω : A → ℂ) (d c : I → A) (r a e : A) | |
| (he : ∀ i,e*c i=c i*e) : | |
| star (assembled d r c)*a*assembled d r c-coefficientValue ω d c a*e= | |
| ∑ i,∑ j,star (c i)*(star r*(star (d i)*a*d j)*r- | |
| ω (star (d i)*a*d j) • e)*c j := by | |
| rw [assembled_square,coefficient_mul ω d c a e he,←Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [←Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| noncomm_ring | |
| theorem assembly_error_le (ω : A → ℂ) (d c : I → A) (r a e : A) | |
| (he : ∀ i,e*c i=c i*e) {δ : ℝ} | |
| (hδ : ∀ i j,‖star r*(star (d i)*a*d j)*r-ω (star (d i)*a*d j) • e‖≤δ) : | |
| ‖star (assembled d r c)*a*assembled d r c-coefficientValue ω d c a*e‖≤ | |
| δ*(∑ i,‖c i‖)^2 := by | |
| rw [assembly_error_identity ω d c r a e he] | |
| calc | |
| ‖∑ i,∑ j,star (c i)*(star r*(star (d i)*a*d j)*r- | |
| ω (star (d i)*a*d j) • e)*c j‖ | |
| ≤∑ i,∑ j,‖c i‖*δ*‖c j‖ := by | |
| apply (norm_sum_le _ _).trans | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| apply (norm_sum_le _ _).trans | |
| apply Finset.sum_le_sum | |
| intro j _ | |
| calc | |
| _≤(‖star (c i)‖*‖star r*(star (d i)*a*d j)*r- | |
| ω (star (d i)*a*d j) • e‖)*‖c j‖ := | |
| (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right | |
| (norm_mul_le _ _) (norm_nonneg _)) | |
| _≤_ := by | |
| rw [norm_star] | |
| exact mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left (hδ i j) (norm_nonneg _)) (norm_nonneg _) | |
| _=δ*(∑ i,‖c i‖)^2 := by | |
| simp only [←Finset.mul_sum,←Finset.sum_mul] | |
| ring | |
| theorem compression_error_le (b r a e : A) (zeta : ℂ) | |
| (hb : star b=b) (hbr : b*r=r) {δ ε : ℝ} | |
| (hδ : ‖b*a*b-zeta • (b*b)‖≤δ) (hε : ‖star r*r-e‖≤ε) : | |
| ‖star r*a*r-zeta • e‖≤‖r‖^2*δ+‖zeta‖*ε := by | |
| have hrb : star r*b=star r := by | |
| have h := congrArg star hbr | |
| simpa only [star_mul,hb] using h | |
| have hid : star r*a*r-zeta • e= | |
| star r*(b*a*b-zeta • (b*b))*r+zeta • (star r*r-e) := by | |
| simp only [mul_sub,sub_mul,mul_smul_comm,smul_mul_assoc,smul_sub] | |
| rw [show star r*(b*a*b)*r=star r*a*r by | |
| calc | |
| star r*(b*a*b)*r=(star r*b)*a*(b*r) := by noncomm_ring | |
| _=star r*a*r := by rw [hrb,hbr]] | |
| rw [show star r*(b*b)*r=star r*r by | |
| calc | |
| star r*(b*b)*r=(star r*b)*(b*r) := by noncomm_ring | |
| _=star r*r := by rw [hrb,hbr]] | |
| abel | |
| rw [hid] | |
| apply (norm_add_le _ _).trans | |
| apply add_le_add | |
| · calc | |
| ‖star r*(b*a*b-zeta • (b*b))*r‖≤ | |
| (‖star r‖*‖b*a*b-zeta • (b*b)‖)*‖r‖ := | |
| (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _)) | |
| _≤(‖r‖*δ)*‖r‖ := by | |
| rw [norm_star] | |
| exact mul_le_mul_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left hδ (norm_nonneg _)) (norm_nonneg _) | |
| _=‖r‖^2*δ := by ring | |
| · rw [norm_smul] | |
| exact mul_le_mul_of_nonneg_left hε (norm_nonneg _) | |
| end CStarExcision | |
| namespace DirectComparison | |
| open Pureness CurrentMain | |
| open scoped ComplexOrder ENNReal | |
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| lemma cut_rank_lower_from_trace (τ : TracialState A) (w : A) | |
| (hw : 0 ≤ w) (hwn : ‖w‖ ≤ 1) {β : ℝ} (hβ : 0 ≤ β) : | |
| (τ.functional w).re-β ≤ (normalizedRootRank τ (directPositiveCut β w)).toReal := by | |
| let b := directPositiveCut β w | |
| have hb := directPositiveCut_nonneg β w | |
| have hb1 : b ≤ 1 := (CStarAlgebra.norm_le_one_iff_of_nonneg b hb).mp | |
| (norm_directPositiveCut_le_one w hw hwn β hβ) | |
| have hr := trace_le_rank_of_mem_hereditary τ b b hb (self_mem_hereditary b hb) hb1 | |
| have hrs := ENNReal.toReal_mono (directRank_ne_top τ b hb) hr | |
| rw [ENNReal.toReal_ofReal (τ.functional.map_nonneg hb).1] at hrs | |
| have he := (trace_sub_real_le_norm τ w b).trans (norm_sub_directPositiveCut_le w hw β hβ) | |
| linarith | |
| theorem small_trace_supported_isometry [Nonempty (TracialState A)] | |
| (hcmp : StrictComparison A) (e w h f : A) | |
| (he : 0 ≤ e) (hen : ‖e‖ ≤ 1) (hw : 0 ≤ w) (hwn : ‖w‖ ≤ 1) (hhw : h*w=w) | |
| {α β ε : ℝ} (hα : 0<α) (hβ : 0<β) (hε : 0<ε) (hε1 : ε ≤ 1) | |
| (hsmall : ∀ τ : TracialState A, (τ.functional e).re<α*β) | |
| (hlarge : ∀ τ : TracialState A, 2*β ≤ (τ.functional w).re) : | |
| ∃ s : A, ‖s‖ ≤ 2 ∧ ‖star s*s-e‖<ε+α ∧ h*s=s ∧ | |
| ‖f*s-s‖ ≤ 2*β⁻¹*‖f*w-w‖ := by | |
| let a := directPositiveCut α e | |
| let b := directPositiveCut β w | |
| have ha := directPositiveCut_nonneg α e | |
| have hb := directPositiveCut_nonneg β w | |
| have hgap (τ : TracialState A) : normalizedRootRank τ a<normalizedRootRank τ b := by | |
| apply (ENNReal.toReal_lt_toReal (directRank_ne_top τ a ha) (directRank_ne_top τ b hb)).mp | |
| have hs := directPositiveCut_rank_le_trace_div τ e he α hα | |
| have hl := cut_rank_lower_from_trace τ w hw hwn hβ.le | |
| have hd : (τ.functional e).re/α<β := (div_lt_iff₀ hα).mpr (by nlinarith [hsmall τ]) | |
| change (normalizedRootRank τ a).toReal ≤ _ at hs | |
| change _ ≤ (normalizedRootRank τ b).toReal at hl | |
| linarith [hlarge τ] | |
| have hb0 : b≠0 := by | |
| intro hz | |
| let τ : TracialState A := Classical.choice inferInstance | |
| have hn := hgap τ | |
| rw [hz,normalizedRootRank_zero] at hn | |
| exact not_lt_of_ge zero_le hn | |
| have hab := hcmp.scalar a b ha hb hb0 hgap | |
| obtain ⟨s,hsn,hse,hhs,hfs⟩ := approximate_controlled_isometry a w h f | |
| (norm_directPositiveCut_le_one e he hen α hα.le) hw hhw hβ hε hε1 hab | |
| refine ⟨s,hsn,?_,hhs,hfs⟩ | |
| have hn := norm_sub_le_norm_sub_add_norm_sub (star s*s) a e | |
| have hc := norm_sub_directPositiveCut_le e he α hα.le | |
| rw [norm_sub_rev e a] at hc | |
| linarith | |
| theorem supported_state_excision [Nonempty (TracialState A)] | |
| (hcmp : StrictComparison A) (e w h f a : A) (z : ℂ) | |
| (he : 0 ≤ e) (hen : ‖e‖ ≤ 1) (hw : 0 ≤ w) (hwn : ‖w‖ ≤ 1) | |
| (hh : IsSelfAdjoint h) (hhw : h*w=w) | |
| {α β ε δ : ℝ} (hα : 0<α) (hβ : 0<β) (hε : 0<ε) (hε1 : ε ≤ 1) | |
| (hδ : 0 ≤ δ) (hex : ‖h*a*h-z • (h*h)‖ ≤ δ) | |
| (hsmall : ∀ τ : TracialState A, (τ.functional e).re<α*β) | |
| (hlarge : ∀ τ : TracialState A, 2*β ≤ (τ.functional w).re) : | |
| ∃ s : A, ‖s‖ ≤ 2 ∧ ‖star s*s-e‖<ε+α ∧ | |
| ‖f*s-s‖ ≤ 2*β⁻¹*‖f*w-w‖ ∧ | |
| ‖star s*a*s-z • e‖ ≤ 4*δ+‖z‖*(ε+α) := by | |
| obtain ⟨s,hsn,hse,hhs,hfs⟩ := small_trace_supported_isometry hcmp e w h f | |
| he hen hw hwn hhw hα hβ hε hε1 hsmall hlarge | |
| refine ⟨s,hsn,hse,hfs,?_⟩ | |
| have hx := CStarExcision.compression_error_le h s a e z hh.star_eq hhs hex hse.le | |
| have hn : ‖s‖^2 ≤ 4 := by nlinarith [norm_nonneg s] | |
| exact hx.trans (add_le_add (mul_le_mul_of_nonneg_right hn hδ) le_rfl) | |
| end DirectComparison | |
| end | |
| end OAI | |