File size: 4,745 Bytes
110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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
|