| import Mathlib |
| import OAI.Analysis.Commutator.ArvesonAnnihilator |
| import OAI.Analysis.Commutator.WOTCompact |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| noncomputable section |
|
|
| open scoped BigOperators InnerProductSpace Pointwise |
|
|
| open ContinuousLinearMapWOT |
|
|
| namespace Kadison.WOTDistance |
|
|
| variable [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| lemma commutant_membership (M : VonNeumannAlgebra H) (T : H →WOT[ℂ] H) : |
| T∈commutant (M : Set (H →L[ℂ] H)) ↔ T.toCLM∈M.commutant := by |
| change (∀ a∈M,ofCLM a*T=T*ofCLM a) ↔ _ |
| rw [VonNeumannAlgebra.mem_commutant_iff] |
| constructor <;> intro h a ha |
| · exact congrArg toCLM (h a ha) |
| · apply toCLM_injective |
| exact h a ha |
|
|
| lemma ball_functional_bound (φ : (H →WOT[ℂ] H) →L[ℂ] ℂ) |
| (hc : 0<c) (hu : 0≤u) |
| (hφ : ∀ T∈normBall (H:=H) c, (φ T).re≤u) : |
| c*‖φ.comp ContinuousLinearMap.WOTofCLM‖≤u := by |
| let ψ : StrongDual ℂ (H →L[ℂ] H) := φ.comp ContinuousLinearMap.WOTofCLM |
| let ψr : StrongDual ℝ (H →L[ℂ] H) := (StrongDual.extendRCLikeₗᵢ (𝕜:=ℂ)).symm ψ |
| have hn : ‖ψr‖=‖ψ‖ := LinearIsometryEquiv.norm_map _ _ |
| have hnorm : ‖ψr‖≤u/c := by |
| apply ContinuousLinearMap.opNorm_le_of_unit_norm (div_nonneg hu hc.le) |
| intro T hT |
| have hp := hφ (ofCLM (c • T)) (by |
| change ‖c • T‖≤c |
| simp [norm_smul,hT,abs_of_pos hc]) |
| have hm := hφ (ofCLM ((-c) • T)) (by |
| change ‖(-c) • T‖≤c |
| simp [norm_smul,hT,abs_of_pos hc]) |
| have hp' : c*ψr T≤u := by |
| change ψr (c • T)≤u at hp |
| simpa only [map_smul,smul_eq_mul] using hp |
| have hm' : -(c*ψr T)≤u := by |
| change ψr ((-c) • T)≤u at hm |
| simpa only [map_smul,smul_eq_mul,neg_mul] using hm |
| rw [Real.norm_eq_abs,abs_le] |
| constructor |
| · apply (le_of_mul_le_mul_left ?_ hc) |
| rw [mul_neg, mul_div_cancel₀ _ hc.ne'] |
| linarith |
| · exact (le_div_iff₀ hc).mpr (by simpa [mul_comm] using hp') |
| rw [hn] at hnorm |
| simpa only [mul_comm] using (le_div_iff₀ hc).mp hnorm |
| |
| theorem exists_commutant_approximation (M : VonNeumannAlgebra H) (G : H →L[ℂ] H) |
| {c : ℝ} (hc : 0≤c) (hcb : AmplifiedBound M G c) : |
| ∃ T : H →L[ℂ] H, T∈M.commutant ∧ ‖G-T‖≤c := by |
| classical |
| let s := (commutant (M : Set (H →L[ℂ] H)) : Set (H →WOT[ℂ] H))+normBall c |
| have hs0 : (0 : H →WOT[ℂ] H)∈s := by |
| exact Set.mem_add.mpr ⟨0,(commutant _).zero_mem,0,by simpa [normBall] using hc,add_zero _⟩ |
| have hs : ofCLM G∈s := by |
| by_contra hG |
| have hsv : Convex ℝ s := ((commutant _).restrictScalars ℝ).convex.add (convex_normBall c) |
| have hsc : IsClosed s := isClosed_commutant_add_ball _ hc |
| obtain ⟨φ,u,hφ,hu⟩ := RCLike.geometric_hahn_banach_closed_point (𝕜:=ℂ) hsv hsc hG |
| have hu0 : 0<u := by simpa using hφ 0 hs0 |
| have hzero : ∀ T∈M.commutant,φ (ofCLM T)=0 := by |
| intro T hT |
| by_contra hn |
| let z : ℂ := ((u+1 : ℝ) : ℂ)/φ (ofCLM T) |
| have hz : z • ofCLM T∈s := by |
| apply Set.mem_add.mpr |
| refine ⟨z • ofCLM T, (commutant _).smul_mem z ?_,0,?_,add_zero _⟩ |
| · exact (commutant_membership M _).mpr hT |
| · simpa [normBall] using hc |
| have hh := hφ (z • ofCLM T) hz |
| rw [map_smul,smul_eq_mul] at hh |
| dsimp [z] at hh |
| rw [div_mul_cancel₀ _ hn,Complex.ofReal_re] at hh |
| linarith |
| have hb : ∀ V∈normBall (H:=H) c,(φ V).re≤u := by |
| intro V hV |
| exact (hφ V (Set.mem_add.mpr ⟨0,(commutant _).zero_mem,V,hV,zero_add _⟩)).le |
| have hnorm := annihilator_bound M G hc hcb φ hzero |
| have hup : c*‖φ.comp ContinuousLinearMap.WOTofCLM‖≤u := by |
| rcases hc.eq_or_lt with h|h |
| · simp [←h,hu0.le] |
| · exact ball_functional_bound φ h hu0.le hb |
| change u < (φ (ofCLM G)).re at hu |
| have hre := Complex.re_le_norm (φ (ofCLM G)) |
| linarith |
| obtain ⟨T,hT,V,hV,hTV⟩ := Set.mem_add.mp hs |
| refine ⟨T.toCLM,(commutant_membership M T).mp hT,?_⟩ |
| have hG : G-T.toCLM=V.toCLM := by |
| have hh := congrArg toCLM hTV |
| change T.toCLM+V.toCLM=G at hh |
| rw [←hh,add_sub_cancel_left] |
| rw [hG] |
| exact hV |
| |
| end Kadison.WOTDistance |
| |
| end |
| |
| end |
| end |
| |
| end OAI |
| |
