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a97875e | 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 | import Mathlib
import OAI.Analysis.Commutator.CFCIntertwiner
import OAI.Analysis.Commutator.CFCTraceMoments
namespace OAI
noncomputable section
section
noncomputable section
open scoped ComplexOrder
namespace Kadison.BalancedDilation
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
lemma sqrt_eq_real_cfc (a : A) (ha : 0≤a) : CFC.sqrt a=cfc Real.sqrt a := by
rw [CFC.sqrt_eq_real_sqrt a ha,cfcₙ_eq_cfc (hf0 := Real.sqrt_zero)]
lemma defect_nonneg (x : A) (hx : ‖x‖≤1) : 0≤1-x*star x := by
apply sub_nonneg.mpr
apply (CStarAlgebra.norm_le_one_iff_of_nonneg (x*star x) (mul_star_self_nonneg x)).mp
rw [CStarRing.norm_self_mul_star]
exact (mul_le_mul_of_nonneg_right hx (norm_nonneg x)).trans (by simpa using hx)
lemma defect_intertwines (x : A) (hx : ‖x‖≤1) :
CFC.sqrt (1-x*star x)*x=x*CFC.sqrt (1-star x*x) := by
have h₁ := defect_nonneg x hx
have h₂ : 0≤1-star x*x := by simpa only [star_star] using defect_nonneg (star x) (by simpa using hx)
rw [sqrt_eq_real_cfc _ h₁,sqrt_eq_real_cfc _ h₂]
exact (CFCIntertwiner.semiconj_cfc x (1-star x*x) (1-x*star x)
(IsSelfAdjoint.of_nonneg h₂) (IsSelfAdjoint.of_nonneg h₁)
(by noncomm_ring) Real.sqrt Real.continuous_sqrt).symm
lemma sqrt_sub_one_cfc (a : A) (ha : IsSelfAdjoint a) (hpos : 0≤1-a) :
CFC.sqrt (1-a)=cfc (fun r : ℝ => Real.sqrt (1-r)) a := by
have he : cfc (fun r : ℝ => 1-r) a=1-a := by
simpa only [cfc_const (1 : ℝ) a ha,map_one,cfc_id' ℝ a ha] using
cfc_sub (fun _ : ℝ => 1) (fun r : ℝ => r) a
rw [sqrt_eq_real_cfc _ hpos,←he,←cfc_comp Real.sqrt (fun r : ℝ => 1-r) a]
rfl
lemma defect_trace (τ : A →ₚ[ℂ] ℂ) (hτ : ∀ a b : A,τ (a*b)=τ (b*a))
(x : A) (hx : ‖x‖≤1) : τ (CFC.sqrt (1-x*star x))=τ (CFC.sqrt (1-star x*x)) := by
have h₁ := defect_nonneg x hx
have h₂ : 0≤1-star x*x := by simpa only [star_star] using defect_nonneg (star x) (by simpa using hx)
rw [sqrt_sub_one_cfc _ (by cfc_tac) h₁,sqrt_sub_one_cfc _ (by cfc_tac) h₂]
apply CFCTraceMoments.trace_cfc_of_powers τ (x*star x) (star x*x)
(by cfc_tac) (by cfc_tac)
(fun n => CFCTraceMoments.trace_cyclic_powers τ hτ x (star x) n)
fun_prop
end Kadison.BalancedDilation
end
end
end
end OAI
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