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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.CompleteAdditivity
namespace OAI
/-! Support projections for completely additive positive functionals. -/
noncomputable section
open Filter Topology
open scoped ComplexOrder
namespace BoundedHochschild
namespace KadisonRingrose
universe u
variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M]
[NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X]
/-- Complete additivity supplies a support projection without assuming
normality in advance. This is used in proving the normality criterion. -/
theorem exists_additive_positive_support (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M)
(ψ : M →ₚ[ℂ] ℂ) (hψ : CompletelyAdditive (positiveFunctionalCLM ψ)) :
∃ p : M, IsStarProjection p ∧ ψ p = ψ 1 ∧
∀ q : M, IsStarProjection q → q ≤ p → ψ q = 0 → q = 0 := by
classical
let A : Set (Set M) := {S | (∀ p ∈ S, IsStarProjection p ∧ p ≠ 0 ∧ ψ p = 0) ∧
∀ p ∈ S, ∀ q ∈ S, p ≠ q → p * q = 0}
obtain ⟨S, hS⟩ := zorn_subset A (fun c hc hchain => by
refine ⟨⋃₀ c, ⟨?_, ?_⟩, fun s hs => Set.subset_sUnion_of_mem hs⟩
· intro p hp
obtain ⟨s, hs, hps⟩ := Set.mem_sUnion.mp hp
exact (hc hs).1 p hps
· intro p hp q hq hpq
obtain ⟨s, hs, hps⟩ := Set.mem_sUnion.mp hp
obtain ⟨t, ht, hqt⟩ := Set.mem_sUnion.mp hq
rcases hchain.total hs ht with hst | hts
· exact (hc ht).2 p (hst hps) q hqt hpq
· exact (hc hs).2 p hps q (hts hqt) hpq)
let f : S → M := Subtype.val
have hf (i : S) : IsStarProjection (f i) := (hS.1.1 i i.property).1
have ho (i j : S) (hij : i ≠ j) : f i * f j = 0 :=
hS.1.2 i i.property j j.property (fun h => hij (Subtype.ext h))
obtain ⟨P, hP, hPu, hPl, _⟩ := exists_orthogonal_projection_sum e f hf ho
have hPnull : ψ P = 0 := by
have ht := hψ S f hf ho P hP hPu hPl
have heq : (fun s : Finset S => ∑ i ∈ s, positiveFunctionalCLM ψ (f i)) =
(fun _ : Finset S => (0 : ℂ)) := by
funext s
apply Finset.sum_eq_zero
intro i _
exact (hS.1.1 i i.property).2.2
rw [heq] at ht
exact tendsto_nhds_unique ht tendsto_const_nhds
refine ⟨1 - P, hP.one_sub, ?_, ?_⟩
· rw [map_sub, hPnull, sub_zero]
· intro q hq hqP hnull
by_contra hq0
have hPq : P * q = 0 :=
HochschildTypeII.projection_orthogonal_of_le_complement hP hq hqP
have hfq (i : S) : f i * q = 0 := by
calc
f i * q = (f i * P) * q := by rw [((hf i).le_iff_mul_eq_left hP).mp (hPu i)]
_ = 0 := by rw [mul_assoc, hPq, mul_zero]
have hqf (i : S) : q * f i = 0 := by
simpa only [star_mul, (hf i).isSelfAdjoint.star_eq, hq.isSelfAdjoint.star_eq,
star_zero] using congrArg star (hfq i)
have hnew : insert q S ∈ A := by
constructor
· intro a ha
rcases Set.mem_insert_iff.mp ha with rfl | ha
· exact ⟨hq, hq0, hnull⟩
· exact hS.1.1 a ha
· intro a ha b hb hab
rcases Set.mem_insert_iff.mp ha with har | ha
· subst a
rcases Set.mem_insert_iff.mp hb with hbr | hb
· exact (hab hbr.symm).elim
· exact hqf ⟨b, hb⟩
· rcases Set.mem_insert_iff.mp hb with hbr | hb
· subst b
exact hfq ⟨a, ha⟩
· exact hS.1.2 a ha b hb hab
have hsubset : insert q S ⊆ S := hS.2 hnew (Set.subset_insert q S)
have hqin : q ∈ S := hsubset (Set.mem_insert q S)
exact hq0 (by simpa only [f, hq.isIdempotentElem.eq] using hfq ⟨q, hqin⟩)
end KadisonRingrose
end BoundedHochschild
end
end OAI
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