| import Mathlib |
| import OAI.Analysis.BoundedHochschild.CocycleNormalParts |
| import OAI.Analysis.BoundedHochschild.FactorFinitePaths |
| import OAI.Analysis.BoundedHochschild.FreeWords |
| import OAI.Analysis.BoundedHochschild.MatrixDisplacement |
| import OAI.Analysis.BoundedHochschild.UnitaryLifts |
| import OAI.Analysis.BoundedHochschild.CentralMatrices |
|
|
| namespace OAI |
|
|
| /-! Central centering and projection displacement. -/ |
|
|
| noncomputable section |
|
|
| open Filter Topology |
| open scoped ComplexOrder |
| namespace HochschildCentralQuantization |
| open BoundedHochschild HochschildComparison HochschildDixmier HochschildRigidity |
| HochschildSakai HochschildQuantization HochschildCentralComparison FinitePathMain |
| variable {M X : Type*} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] |
| [NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X] |
| variable (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) (τ : M →ₚ[ℂ] ℂ) |
|
|
| def CentrallyCentered (a : M) : Prop := ∀ z : M, (∀ x : M, Commute z x) → τ (z*a)=0 |
|
|
| omit [PartialOrder M] [StarOrderedRing M] in |
| lemma central_star {b : M} (hb : ∀ x : M, Commute b x) : ∀ x : M, Commute (star b) x := by |
| intro x |
| have hh := congrArg star (hb (star x)).eq |
| change star b*x=x*star b |
| simpa only [star_mul,star_star] using hh.symm |
|
|
| include e in |
| lemma trace_weighted_orbit_closure |
| (hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) |
| (a b z : M) (hz : ∀ x : M, Commute z x) |
| (hb : gnsVector τ b ∈ closure (gnsVector τ '' convexHull ℝ |
| (Set.range (fun u : unitary M => conjugate u a)))) : τ (z*b)=τ (z*a) := by |
| obtain ⟨c,hc,hcn,hct⟩ := orbit_sequence τ hb |
| have ht := trace_predual_tendsto e τ hτ (predual_tendsto_mul_left e z |
| (predual_tendsto_of_preGNS e τ (mul_hasPreadjoint e) hτ hfaith ‖a‖ hcn hct)) |
| have he : ∀ n, τ (z*c n)=τ (z*a) := fun n => central_trace_orbit τ htr a z hz (hc n) |
| apply tendsto_nhds_unique ht |
| simpa only [he] using (tendsto_const_nhds : Tendsto (fun _ : ℕ => τ (z*a)) atTop (𝓝 _)) |
|
|
| include e in |
| lemma zero_mem_orbit_closure |
| (hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) |
| {a : M} (ha : CentrallyCentered τ a) : |
| gnsVector τ (0:M) ∈ closure (gnsVector τ '' convexHull ℝ |
| (Set.range (fun u : unitary M => conjugate u a))) := by |
| obtain ⟨b,_hbn,hbc,hb⟩ := exists_central_average e τ (mul_hasPreadjoint e) hτ htr hfaith a |
| have hb0 : b=0 := by |
| apply hfaith |
| rw [trace_weighted_orbit_closure e τ hτ htr hfaith a b (star b) (central_star hbc) hb] |
| exact ha (star b) (central_star hbc) |
| simpa only [hb0] using hb |
|
|
| include e in |
| lemma norm_le_of_conjugations |
| (hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) |
| {a : M} (ha : CentrallyCentered τ a) (ε : ℝ) |
| (hu : ∀ u : unitary M, ‖τ.toPreGNS (conjugate u a-a)‖ ≤ ε) : |
| ‖τ.toPreGNS a‖ ≤ ε := by |
| have hk : gnsVector τ '' convexHull ℝ |
| (Set.range (fun u : unitary M => conjugate u a)) ⊆ |
| Metric.closedBall (gnsVector τ a) ε := by |
| rw [LinearMap.image_convexHull] |
| apply convexHull_min _ (convex_closedBall _ _) |
| rintro _ ⟨_,⟨u,rfl⟩,rfl⟩ |
| simpa only [Metric.mem_closedBall,dist_eq_norm,← map_sub,gnsVector_norm] using hu u |
| have hh := closure_minimal hk Metric.isClosed_closedBall |
| (zero_mem_orbit_closure e τ hτ htr hfaith ha) |
| simpa only [Metric.mem_closedBall,dist_eq_norm,map_zero,zero_sub,norm_neg,gnsVector_norm] using hh |
|
|
| include e in |
| lemma norm_le_of_projection_commutators |
| (hτ : TraceInPredual e τ) (h1 : τ 1=1) |
| (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) |
| {a : M} (ha : CentrallyCentered τ a) (K : ℝ) (hK : 0 ≤ K) |
| (hp : ∀ p : M, IsStarProjection p → ‖τ.toPreGNS (p*a-a*p)‖ ≤ K) : |
| ‖τ.toPreGNS a‖ ≤ 4*K := by |
| apply norm_le_of_conjugations e τ hτ htr hfaith ha (4*K) |
| intro u |
| have : Nontrivial M := ⟨⟨1,0,by |
| intro hz |
| have hh : (1:ℂ)=0 := h1.symm.trans ((congrArg τ hz).trans (map_zero τ)) |
| exact one_ne_zero hh⟩⟩ |
| have hh := contraction_bound_of_projections e (commutatorGNS τ a) (2*‖a‖) K |
| (by positivity) hK (commutatorGNS_bound τ h1 a) hp (u:M) |
| (CStarRing.norm_coe_unitary u).le |
| have he : conjugate u a-a=((u:M)*a-a*(u:M))*star (u:M) := by |
| simp only [conjugate_apply,sub_mul,mul_assoc,Unitary.mul_star_self_of_mem u.prop,mul_one] |
| rw [he,show star (u:M)=((star u:unitary M):M) from rfl,preGNS_norm_unitary_right τ htr] |
| exact hh |
|
|
| include e in |
| lemma exists_central_projection_displacement |
| (hτ : TraceInPredual e τ) (h1 : τ 1=1) |
| (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) |
| {a : M} (ha : CentrallyCentered τ a) (ha0 : 0<‖τ.toPreGNS a‖) : |
| ∃ p : M, IsStarProjection p ∧ ‖τ.toPreGNS a‖/8 < ‖τ.toPreGNS (p*a-a*p)‖ := by |
| by_contra hn |
| push Not at hn |
| have hh := norm_le_of_projection_commutators e τ hτ h1 htr hfaith ha |
| (‖τ.toPreGNS a‖/8) (by positivity) hn |
| linarith |
|
|
| end HochschildCentralQuantization |
|
|
| end |
|
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| end OAI |
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