| import OAI.Analysis.BoundedHochschild.CentralSupportNormalizedRows | |
| import OAI.Analysis.BoundedHochschild.InfiniteRepresentedFamilies | |
| namespace OAI | |
| /-! The row exhausting a central support compresses the averaging operator | |
| through the original projection corner. -/ | |
| noncomputable section | |
| open Filter Topology | |
| namespace BoundedHochschild.KadisonRingrose | |
| open HochschildComparison | |
| variable {M H K : Type*} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] | |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K] | |
| lemma centralSupportRow_compression_term | |
| (π : M →⋆ₐ[ℂ] (K →L[ℂ] K)) (σ : M →⋆ₐ[ℂ] (H →L[ℂ] H)) | |
| {p z : M} (hp : IsStarProjection p) (hz : IsStarProjection z) (hpz : p ≤ z) | |
| (S : Set (PartialIso M)) (hS : ∀ v ∈ S, v.final ≤ p ∧ v.initial ≤ z) | |
| (W : H →L[ℂ] K) (i : Option S) (ξ : H) : | |
| π (centralSupportRow z S i) (W ((σ (centralSupportRow z S i)).adjoint (σ z ξ))) = | |
| π (centralSupportRow z S i) | |
| (π p (W (σ p ((σ (centralSupportRow z S i)).adjoint ξ)))) := by | |
| cases i with | |
| | none => | |
| have hright : (σ (1 - z)).adjoint.comp (σ z) = 0 := by | |
| rw [← ContinuousLinearMap.star_eq_adjoint, ← map_star, | |
| hz.one_sub.isSelfAdjoint.star_eq] | |
| change σ (1 - z) * σ z = 0 | |
| rw [← map_mul, hz.one_sub_mul_self, map_zero] | |
| have hleft : (π (1 - z)).comp (π p) = 0 := by | |
| change π (1 - z) * π p = 0 | |
| rw [← map_mul, sub_mul, one_mul, (hp.le_iff_mul_eq_right hz).mp hpz, | |
| sub_self, map_zero] | |
| have hr := congrArg (fun A : H →L[ℂ] H => A ξ) hright | |
| have hl := congrArg (fun A : K →L[ℂ] K => A (W (σ p ((σ (1-z)).adjoint ξ)))) hleft | |
| change (σ (1 - z)).adjoint (σ z ξ) = 0 at hr | |
| change π (1 - z) (π p (W (σ p ((σ (1-z)).adjoint ξ)))) = 0 at hl | |
| change π (1 - z) (W ((σ (1-z)).adjoint (σ z ξ))) = _ | |
| rw [hr, map_zero, map_zero] | |
| exact hl.symm | |
| | some v => | |
| have hpv : p * (v.val : M) = v.val := by | |
| calc | |
| _ = p * (v.val.final * (v.val : M)) := by rw [v.val.final_mul] | |
| _ = v.val := by rw [← mul_assoc, | |
| (v.val.final_proj.le_iff_mul_eq_right hp).mp (hS v v.property).1, | |
| v.val.final_mul] | |
| have hvz : (v.val : M) * z = v.val := by | |
| calc | |
| _ = ((v.val : M) * v.val.initial) * z := by rw [v.val.mul_initial] | |
| _ = v.val := by rw [mul_assoc, | |
| (v.val.initial_proj.le_iff_mul_eq_left hz).mp (hS v v.property).2, | |
| v.val.mul_initial] | |
| have hvp : star (v.val : M) * p = star (v.val : M) := by | |
| simpa only [star_mul, hp.isSelfAdjoint.star_eq] using congrArg star hpv | |
| have hσ (a b c : M) (h : a * b = c) (η : H) : σ a (σ b η) = σ c η := | |
| congrArg (fun A : H →L[ℂ] H => A η) ((map_mul σ a b).symm.trans (congrArg σ h)) | |
| have hπ (a b c : M) (h : a * b = c) (η : K) : π a (π b η) = π c η := | |
| congrArg (fun A : K →L[ℂ] K => A η) ((map_mul π a b).symm.trans (congrArg π h)) | |
| have hs : (σ (star (v.val : M))).adjoint = σ (v.val : M) := by | |
| rw [← ContinuousLinearMap.star_eq_adjoint, ← map_star, star_star] | |
| simp only [centralSupportRow, hs, hσ _ _ _ hvz, hσ _ _ _ hpv, hπ _ _ _ hvp] | |
| theorem centralSupportRow_compression_norm_le | |
| (π : M →⋆ₐ[ℂ] (K →L[ℂ] K)) (σ : M →⋆ₐ[ℂ] (H →L[ℂ] H)) | |
| {p z : M} (hp : IsStarProjection p) (hz : IsStarProjection z) (hpz : p ≤ z) | |
| (S : Set (PartialIso M)) (hS : ∀ v ∈ S, v.final ≤ p ∧ v.initial ≤ z) | |
| (hm : ∀ s : Finset (Option S), | |
| ∑ i ∈ s, centralSupportRow z S i * star (centralSupportRow z S i) ≤ 1) | |
| (W : H →L[ℂ] K) : | |
| ‖(infiniteRepresentedRightAverage π σ (centralSupportRow z S) hm W).comp (σ z)‖ ≤ | |
| ‖(π p).comp (W.comp (σ p))‖ := by | |
| have he : (infiniteRepresentedRightAverage π σ (centralSupportRow z S) hm W).comp (σ z) = | |
| infiniteRepresentedRightAverage π σ (centralSupportRow z S) hm | |
| ((π p).comp (W.comp (σ p))) := by | |
| ext ξ | |
| have h₁ := infiniteRepresentedRightAverage_hasSum π σ (centralSupportRow z S) hm W (σ z ξ) | |
| have h₂ := infiniteRepresentedRightAverage_hasSum π σ (centralSupportRow z S) hm | |
| ((π p).comp (W.comp (σ p))) ξ | |
| have hf := funext (fun i => centralSupportRow_compression_term π σ hp hz hpz S hS W i ξ) | |
| simp only [ContinuousLinearMap.comp_apply] at h₂ ⊢ | |
| rw [hf] at h₁ | |
| exact h₁.unique h₂ | |
| rw [he] | |
| exact infiniteRepresentedRightAverage_norm_le π σ (centralSupportRow z S) hm _ | |
| end BoundedHochschild.KadisonRingrose | |
| end | |
| end OAI | |