| import OAI.Analysis.BoundedHochschild.CompleteAdditivity |
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| namespace OAI |
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| /-! Support projections for completely additive positive functionals. -/ |
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| noncomputable section |
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| open Filter Topology |
| open scoped ComplexOrder |
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| namespace BoundedHochschild |
| namespace KadisonRingrose |
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| universe u |
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| variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] |
| [NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X] |
|
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| /-- 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⟩) |
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| end KadisonRingrose |
| end BoundedHochschild |
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| end |
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| end OAI |
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