| import OAI.Analysis.BoundedHochschild.CentralSupportRows |
|
|
| namespace OAI |
|
|
| /-! Complete the row covering a central support by the complementary central |
| projection. The resulting row has square sum one in the given predual. -/ |
|
|
| noncomputable section |
|
|
| open Filter Topology |
| open scoped ComplexOrder |
|
|
| namespace BoundedHochschild.KadisonRingrose |
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|
| open HochschildComparison |
|
|
| universe u |
| variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] |
| [NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X] |
|
|
| def centralSupportRow (z : M) (S : Set (PartialIso M)) : Option S → M |
| | none => 1 - z |
| | some v => star (v.val : M) |
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|
| omit [PartialOrder M] [StarOrderedRing M] in |
| lemma centralSupportRow_square (z : M) (hz : IsStarProjection z) |
| (S : Set (PartialIso M)) (i : Option S) : |
| centralSupportRow z S i * star (centralSupportRow z S i) = |
| Option.elim i (1 - z) (fun v => v.val.initial) := by |
| cases i with |
| | none => |
| change (1 - z) * star (1 - z) = 1 - z |
| rw [hz.one_sub.isSelfAdjoint.star_eq, hz.one_sub.isIdempotentElem.eq] |
| | some v => simp only [centralSupportRow, star_star, PartialIso.initial, Option.elim_some] |
|
|
| theorem centralSupportRow_normalized (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) |
| {z : M} (hz : IsStarProjection z) (S : Set (PartialIso M)) |
| (hvz : ∀ v ∈ S, v.initial ≤ z) |
| (horth : ∀ v ∈ S, ∀ w ∈ S, v ≠ w → v.initial * w.initial = 0) |
| (ht : PredualTendsto e (fun s : Finset S => ∑ v ∈ s, v.val.initial) atTop z) : |
| (∀ s : Finset (Option S), |
| ∑ i ∈ s, centralSupportRow z S i * star (centralSupportRow z S i) ≤ 1) ∧ |
| PredualTendsto e (fun s : Finset (Option S) => |
| ∑ i ∈ s, centralSupportRow z S i * star (centralSupportRow z S i)) atTop 1 := by |
| classical |
| let q : Option S → M := fun i => Option.elim i (1 - z) (fun v => v.val.initial) |
| have hq (i : Option S) : IsStarProjection (q i) := by |
| cases i with |
| | none => exact hz.one_sub |
| | some v => exact v.val.initial_proj |
| have ho (i j : Option S) (hij : i ≠ j) : q i * q j = 0 := by |
| cases i with |
| | none => |
| cases j with |
| | none => exact (hij rfl).elim |
| | some v => |
| change (1 - z) * v.val.initial = 0 |
| rw [sub_mul, one_mul, (v.val.initial_proj.le_iff_mul_eq_right hz).mp |
| (hvz v v.property), sub_self] |
| | some v => |
| cases j with |
| | none => |
| change v.val.initial * (1 - z) = 0 |
| rw [mul_sub, mul_one, (v.val.initial_proj.le_iff_mul_eq_left hz).mp |
| (hvz v v.property), sub_self] |
| | some w => |
| exact horth v v.property w w.property |
| (fun h => hij (congrArg some (Subtype.ext h))) |
| obtain ⟨Q, hQ, hiQ, _, htQ⟩ := exists_orthogonal_projection_sum e q hq ho |
| have hzQ : z ≤ Q := by |
| apply (hz.le_iff_mul_eq_left hQ).mpr |
| have hh := orthogonal_projection_sum_annihilator e (fun v : S => v.val.initial) |
| (1 - Q) z ht (fun v => by |
| rw [mul_sub, mul_one, (v.val.initial_proj.le_iff_mul_eq_left hQ).mp |
| (hiQ (some v)), sub_self]) |
| rw [mul_sub, mul_one, sub_eq_zero] at hh |
| exact hh.symm |
| have hQ1 : Q = 1 := by |
| have hzfix := (hz.le_iff_mul_eq_right hQ).mp hzQ |
| have hcfix := (hz.one_sub.le_iff_mul_eq_right hQ).mp (hiQ none) |
| have he : Q * (z + (1 - z)) = z + (1 - z) := by rw [mul_add, hzfix, hcfix] |
| simpa only [add_sub_cancel, mul_one] using he |
| have he (i : Option S) := centralSupportRow_square z hz S i |
| constructor |
| · intro s |
| simpa only [he] using (orthogonal_projection_finset_sum q hq ho s).le_one |
| · simpa only [he, hQ1] using htQ |
|
|
| end BoundedHochschild.KadisonRingrose |
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|
| end |
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| end OAI |
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|
