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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 | 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
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)
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
end
end OAI
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