| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.MatrixAlgebras.FamilyProjectionLiftTraces |
| import OAI.Analysis.JiangSu.Excision.PureStateSmallExcision |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace NonunitalSI |
| open Pureness CurrentMain NonunitalGamma PureStateExcision Filter Topology CStarAlgebra |
| open scoped ComplexOrder ENNReal |
|
|
| variable {A : Type} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable [Nontrivial A] [Nonempty (NormalizedTrace A)] [CompactSpace (NormalizedTrace A)] [TopologicalSpace.SeparableSpace A] |
| variable (U : Ultrafilter ℕ) |
| local notation "N" => NormUltra.QuotientAlgebra (fun _ : ℕ => A) U |
| local notation "T" => (fun τ : NormalizedTrace A => NormalizedTrace.unitized τ) |
| local notation "q" => normToTracial (A := A) U |
| local notation "L" => familyLimitTrace T U |
|
|
| theorem pure_state_excision_of_trace_projection |
| (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hprojectionless : ∀ p : A,IsStarProjection p → p=0) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) |
| (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 → NormalizedTrace A,2*t ≤ ((L s).functional (q F)).re) |
| (φ : Unitization ℂ A →ₚ[ℂ] ℂ) (hφ1 : φ 1=1) |
| (hpure : ofPositiveState φ hφ1∈(realStateCarrier (A := Unitization ℂ A)).extremePoints ℝ) : |
| ∃ z : N,‖z‖≤1 ∧ star z*z=E ∧ F*z=z ∧ |
| ∀ a : A,star z*NonunitalGamma.normInclusion U a*z=φ (Unitization.inr 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*boundedConstant (A := A) a) (boundedConstant (A := A) 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 : NormalizedTrace A, |
| (τ.functional (e n)).re<η := by |
| have hezero : (familyNullIdeal T (U : Filter ℕ)).quotientMap (boundedInr (A := A) e)=0 := by |
| change q (NormUltra.QuotientAlgebra.mk U e)=0 |
| rwa [heq] |
| have hz := (familyQuotientMap_eq_zero T U (boundedInr (A := A) e)).mp hezero |
| filter_upwards [hz.eventually_lt_const hη] with n hn τ |
| rw [←NormalizedTrace.unitized_inr] |
| exact ((Complex.re_le_norm _).trans ((trace_norm_le_twoNorm (NormalizedTrace.unitized τ) _).trans |
| (traceTwoNorm_le_family T τ _))).trans_lt hn |
| have hpr : IsStarProjection ((familyNullIdeal T (U : Filter ℕ)).quotientMap (boundedInr (A := A) f)) := by |
| change IsStarProjection (q (NormUltra.QuotientAlgebra.mk U f)) |
| rwa [hfq] |
| have hbig : ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A,t≤(τ.functional (f n)).re := by |
| have hg := FamilyProjection.uniform_trace_gap_eventually T U |
| (0 : lp (fun _ : ℕ => Unitization ℂ A) ∞) (boundedInr (A := A) f) ht (by |
| intro s |
| have hh : (familyNullIdeal T (U : Filter ℕ)).quotientStarHom (boundedInr (A := A) f)=q F := by |
| change q (NormUltra.QuotientAlgebra.mk U f)=q F |
| rw [hfq] |
| simpa only [map_zero,Complex.zero_re,zero_add,hh] using hlarge s) |
| simpa only [lp.coeFn_zero,Pi.zero_apply,map_zero,Complex.zero_re,zero_add, |
| boundedInr_apply,NormalizedTrace.unitized_inr] using hg |
| obtain ⟨z,hzn,hze,hzf,hza⟩ := pure_state_excision_norm_ultrapower hsimple hprojectionless hcone hcmp U hfree |
| e f hep hen' hfp hfn' hcen hsm ht hbig |
| (fun η hη => by |
| simpa only [boundedInr_apply,←Unitization.inr_mul,←Unitization.inr_sub, |
| NormalizedTrace.unitized_inr] using |
| FamilyProjection.projection_lift_defect T U (boundedInr (A := A) 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 NonunitalSI |
|
|
| end |
| end |
|
|
| end OAI |
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|
