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a458723 | 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 | import Mathlib
namespace OAI
noncomputable section
section
noncomputable section
open scoped BoundedContinuousFunction
namespace Kadison.BoundedPositiveSequence
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
variable {ι : Type*} [TopologicalSpace ι] [DiscreteTopology ι]
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)]
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)
end Kadison.BoundedPositiveSequence
end
end
end
end OAI
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