| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| open scoped BoundedContinuousFunction |
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| namespace Kadison.BoundedPositiveSequence |
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| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
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| variable {ι : Type*} [TopologicalSpace ι] [DiscreteTopology ι] |
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| theorem squareRoot (d : ι →ᵇ A) (hd : ∀ i, 0≤d i) : |
| ∃ z : ι →ᵇ A, d=star z*z := by |
| let z : ι →ᵇ A := BoundedContinuousFunction.ofNormedAddCommGroupDiscrete |
| (fun i => CFC.sqrt (d i)) (Real.sqrt ‖d‖) (fun i => by |
| rw [CFC.norm_sqrt _ (hd i)] |
| exact Real.sqrt_le_sqrt (d.norm_coe_le_norm i)) |
| refine ⟨z,?_⟩ |
| apply BoundedContinuousFunction.ext |
| intro i |
| change d i=star (CFC.sqrt (d i))*CFC.sqrt (d i) |
| rw [(CFC.sqrt_nonneg (d i)).star_eq,CFC.sqrt_mul_sqrt_self _ (hd i)] |
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| lemma map_nonneg {B : Type*} [CStarAlgebra B] [PartialOrder B] [StarOrderedRing B] |
| (φ : (ι →ᵇ A) →⋆ₐ[ℂ] B) (d : ι →ᵇ A) (hd : ∀ i, 0≤d i) : 0≤φ d := by |
| obtain ⟨z,rfl⟩ := squareRoot d hd |
| simpa only [map_mul,map_star] using star_mul_self_nonneg (φ z) |
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| end Kadison.BoundedPositiveSequence |
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| end |
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| end |
| end |
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| end OAI |
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