| import Mathlib |
| import OAI.Analysis.Commutator.ArvesonCorner |
| import OAI.Analysis.Commutator.CpUltralimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
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| section |
|
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| noncomputable section |
|
|
| open scoped BigOperators InnerProductSpace ComplexOrder CStarAlgebra Topology |
|
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| open Filter |
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| namespace Kadison.OperatorSystem |
|
|
| variable {A H : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| theorem exists_extension (S : Submodule ℂ A) (h1 : (1 : A)∈S) |
| (hs : ∀ a∈S,star a∈S) (f : S →ₗ[ℂ] (H →L[ℂ] H)) |
| (hf : IsCompletelyPositive S f) : |
| ∃ Φ : A →CP (H →L[ℂ] H),∀ a : S,Φ a=f a := by |
| classical |
| have he (t : Finset H) : ∃ Φ : A →CP (H →L[ℂ] H), |
| (∀ a,‖Φ a‖≤‖f ⟨1,h1⟩‖*‖a‖) ∧ |
| ∀ a : S,∀ x∈t,∀ y∈t,⟪Φ a x,y⟫_ℂ=⟪f a x,y⟫_ℂ := by |
| let E := Submodule.span ℂ (t : Set H) |
| let : FiniteDimensional ℂ E := FiniteDimensional.span_of_finite ℂ t.finite_toSet |
| obtain ⟨Φ,hb,hf⟩ := exists_corner_extension S h1 hs f hf E |
| exact ⟨Φ,hb,fun a x hx y hy => hf a x (Submodule.subset_span hx) y (Submodule.subset_span hy)⟩ |
| choose φ hφ hext using he |
| let ω := Ultrafilter.of (atTop : Filter (Finset H)) |
| obtain ⟨Φ,hΦ,hlim⟩ := CpOperations.exists_cp_ultralimit ω φ ‖f ⟨1,h1⟩‖ (norm_nonneg _) hφ |
| refine ⟨Φ,?_⟩ |
| intro a |
| ext x |
| apply ext_inner_right ℂ |
| intro y |
| apply tendsto_nhds_unique (hlim a x y) |
| have hev : ∀ᶠ t in (ω : Filter (Finset H)),⟪φ t a x,y⟫_ℂ=⟪f a x,y⟫_ℂ := by |
| apply (Ultrafilter.of_le _) |
| filter_upwards [eventually_ge_atTop ({x,y} : Finset H)] with t ht |
| exact hext t a x (ht (by simp)) y (ht (by simp)) |
| exact tendsto_const_nhds.congr' (Filter.EventuallyEq.symm hev) |
|
|
| end Kadison.OperatorSystem |
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| end |
|
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| end |
| end |
|
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| end OAI |
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|
