| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.Traces.UnitalNormToTracial |
| import OAI.Analysis.JiangSu.MatrixAlgebras.ProjectionLiftTraces |
| import OAI.Analysis.JiangSu.Excision.PureStateExcision |
| import OAI.Analysis.JiangSu.CentralSequences.NormUltrapowerEpsilonTest |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace DirectComparison |
| open Pureness CurrentMain PureStateExcision Filter Topology CStarAlgebra |
| open scoped ComplexOrder ENNReal |
|
|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| theorem pure_state_excision_norm_ultrapower [Nonempty (TracialState A)] [TopologicalSpace.SeparableSpace A] |
| (hsimple : TopologicallySimple A) (hcmp : StrictComparison A) (U : Ultrafilter ℕ) |
| (hfree : (U : Filter ℕ) ≤ atTop) |
| (e f : lp (fun _ : ℕ => A) ∞) |
| (he : ∀ n,0 ≤ e n) (hen : ∀ n,‖e n‖ ≤ 1) |
| (hf : ∀ n,0 ≤ f n) (hfn : ∀ n,‖f n‖ ≤ 1) |
| (hcentral : ∀ a : A,Tendsto (fun n => ‖f n*a-a*f n‖) (U : Filter ℕ) (𝓝 0)) |
| (hsmall : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A, |
| (τ.functional (e n)).re<η) |
| {t : ℝ} (ht : 0<t) |
| (hlarge : ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A,t ≤ (τ.functional (f n)).re) |
| (hdefect : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A, |
| (τ.functional (f n-f n*f n)).re<η) |
| (φ : A →ₚ[ℂ] ℂ) (hφ1 : φ 1=1) |
| (hpure : ofPositiveState φ hφ1∈(realStateCarrier (A := A)).extremePoints ℝ) |
| : ∃ s : NormUltra.QuotientAlgebra (fun _ : ℕ => A) U, ‖s‖ ≤ 1 ∧ |
| star s*s=NormUltra.QuotientAlgebra.mk U e ∧ |
| NormUltra.QuotientAlgebra.mk U f*s=s ∧ |
| ∀ a : A,star s*NonunitalGamma.normInclusion U a*s= |
| φ a • NormUltra.QuotientAlgebra.mk U e := by |
| classical |
| obtain ⟨a,ha⟩ := TopologicalSpace.exists_dense_seq A |
| let g : ℕ → ℕ → A → ℝ |
| | 0,k,s => ‖star s*s-e k‖ |
| | 1,k,s => ‖f k*s-s‖ |
| | j+2,k,s => ‖star s*a j*s-φ (a j) • e k‖ |
| have hg (j k : ℕ) (s : A) : 0 ≤ g j k s := by |
| rcases j with _|(_|j) <;> exact norm_nonneg _ |
| have hsupply (m : ℕ) (ε : ℝ) (hε : 0<ε) : |
| ∀ᶠ k in (U : Filter ℕ),∃ s : A,‖s‖ ≤ 2 ∧ ∀ j ≤ m,g j k s<ε := by |
| have hs := pure_state_small_excision hsimple hcmp U e f he hen hf hfn hcentral |
| hsmall ht hlarge hdefect φ hφ1 hpure (fun i : Fin (m+1) => a i) hε |
| filter_upwards [hs] with k hk |
| obtain ⟨s,hsn,hse,hsf,hsa⟩ := hk |
| refine ⟨s,hsn,?_⟩ |
| intro j hj |
| rcases j with _|(_|j) |
| · exact hse |
| · exact hsf |
| · exact hsa ⟨j,by omega⟩ |
| obtain ⟨z,hzn,hzg⟩ := NormUltra.bounded_epsilon_test U hfree 2 (by norm_num) g hg hsupply |
| let q := NormUltra.QuotientAlgebra.mkHom (A := fun _ : ℕ => A) (U := U) |
| have hse : star (q z)*q z=q e := |
| (NormUltra.QuotientAlgebra.mk_eq_mk_iff (star z*z) e).mpr (hzg 0) |
| have hsf : q f*q z=q z := |
| (NormUltra.QuotientAlgebra.mk_eq_mk_iff (f*z) z).mpr (hzg 1) |
| have hcoeff (i : ℕ) : star (q z)*NonunitalGamma.normInclusion U (a i)*q z= |
| φ (a i) • q e := |
| (NormUltra.QuotientAlgebra.mk_eq_mk_iff (star z*boundedDiagonal (a i)*z) |
| (φ (a i) • e)).mpr (hzg (i+2)) |
| have hnorme : ‖q e‖ ≤ 1 := (NormUltra.QuotientAlgebra.norm_mk_le e).trans |
| (lp.norm_le_of_forall_le zero_le_one hen) |
| have hnorm : ‖q z‖ ≤ 1 := by |
| have hh : ‖star (q z)*q z‖=‖q z‖*‖q z‖ := CStarRing.norm_star_mul_self |
| rw [hse] at hh |
| nlinarith [norm_nonneg (q z)] |
| refine ⟨q z,hnorm,hse,hsf,?_⟩ |
| have hc1 : Continuous (fun b : A => star (q z)*NonunitalGamma.normInclusion U b*q z) := by fun_prop |
| have hc2 : Continuous (fun b : A => φ b • q e) := by fun_prop |
| have hge := ha.equalizer hc1 hc2 (funext hcoeff) |
| exact fun b => congrFun hge b |
|
|
| end DirectComparison |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace DirectComparison |
| open Pureness CurrentMain PureStateExcision Filter Topology CStarAlgebra |
| open scoped ComplexOrder ENNReal |
|
|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable [Nontrivial A] [Nonempty (TracialState A)] [TopologicalSpace.SeparableSpace A] |
| variable (U : Ultrafilter ℕ) |
| local notation "N" => NormUltra.QuotientAlgebra (fun _ : ℕ => A) U |
| local notation "T" => (fun τ : TracialState A => τ) |
| local notation "q" => normToTracial (A := A) U |
| local notation "L" => familyLimitTrace T U |
|
|
| theorem pure_state_excision_of_trace_projection |
| (hsimple : TopologicallySimple A) (hcmp : StrictComparison A) |
| (hfree : (U : Filter ℕ) ≤ atTop) |
| (E F : N) (he : 0≤E) (hen : ‖E‖≤1) (hsmall : q E=0) |
| (hf : 0≤F) (hfn : ‖F‖≤1) |
| (hcentral : ∀ a : A,F*NonunitalGamma.normInclusion U a=NonunitalGamma.normInclusion U a*F) |
| (hprojection : IsStarProjection (q F)) |
| {t : ℝ} (ht : 0<t) (hlarge : ∀ s : ℕ → TracialState A,2*t ≤ ((L s).functional (q F)).re) |
| (φ : A →ₚ[ℂ] ℂ) (hφ1 : φ 1=1) |
| (hpure : ofPositiveState φ hφ1∈(realStateCarrier (A := A)).extremePoints ℝ) : |
| ∃ z : N,‖z‖≤1 ∧ star z*z=E ∧ F*z=z ∧ |
| ∀ a : A,star z*NonunitalGamma.normInclusion U a*z=φ a • E := by |
| classical |
| obtain ⟨e,heq,hep,hen'⟩ := NormUltra.QuotientAlgebra.positive_contraction_representative he hen |
| obtain ⟨f,hfq,hfp,hfn'⟩ := NormUltra.QuotientAlgebra.positive_contraction_representative hf hfn |
| have hcen (a : A) : Tendsto (fun n => ‖f n*a-a*f n‖) (U : Filter ℕ) (𝓝 0) := by |
| apply (NormUltra.QuotientAlgebra.mk_eq_mk_iff (f*boundedDiagonal a) (boundedDiagonal a*f)).mp |
| rw [NormUltra.QuotientAlgebra.mk_mul,NormUltra.QuotientAlgebra.mk_mul,hfq] |
| exact hcentral a |
| have hsm (η : ℝ) (hη : 0<η) : ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A, |
| (τ.functional (e n)).re<η := by |
| have hezero : (familyNullIdeal T (U : Filter ℕ)).quotientMap e=0 := by |
| change q (NormUltra.QuotientAlgebra.mk U e)=0 |
| rwa [heq] |
| have hz := (familyQuotientMap_eq_zero T U e).mp hezero |
| filter_upwards [hz.eventually_lt_const hη] with n hn τ |
| exact ((Complex.re_le_norm _).trans ((trace_norm_le_twoNorm τ _).trans |
| (traceTwoNorm_le_family T τ _))).trans_lt hn |
| have hpr : IsStarProjection ((familyNullIdeal T (U : Filter ℕ)).quotientMap f) := by |
| change IsStarProjection (q (NormUltra.QuotientAlgebra.mk U f)) |
| rwa [hfq] |
| have hbig : ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A,t≤(τ.functional (f n)).re := by |
| have hg := uniform_trace_gap_eventually U (0 : lp (fun _ : ℕ => A) ∞) f ht (by |
| intro s |
| simpa only [map_zero,Complex.zero_re,zero_add,←normToTracial_mk,hfq] using hlarge s) |
| change ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A, |
| (τ.functional (0:A)).re+t≤(τ.functional (f n)).re at hg |
| simpa only [map_zero,Complex.zero_re,zero_add] using hg |
| obtain ⟨z,hzn,hze,hzf,hza⟩ := pure_state_excision_norm_ultrapower hsimple hcmp U hfree |
| e f hep hen' hfp hfn' hcen hsm ht hbig |
| (fun η hη => projection_lift_defect U f hpr hη) φ hφ1 hpure |
| exact ⟨z,hzn,by rwa [heq] at hze,by rwa [hfq] at hzf, |
| fun a => by simpa only [heq] using hza a⟩ |
|
|
| end DirectComparison |
|
|
| end |
| end |
|
|
| end OAI |
|
|
