| 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 |
|
|
