| import OAI.Analysis.IrrationalRotation.Spectral.TracialInvertibleDeterminant |
| import OAI.Analysis.IrrationalRotation.Spectral.QuasinilpotentDeterminantReduction |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology ENNReal CStarAlgebra |
| namespace Invariant233 |
|
|
| variable {A : Type*} [CStarAlgebra A] |
|
|
| theorem continuousOn_log_normSquare : |
| ContinuousOn (fun a : A => CFC.log (star a * a)) {a : A | IsUnit a} := by |
| apply CFC.continuousOn_log.comp (continuous_star.mul continuous_id).continuousOn |
| intro a ha |
| exact ⟨by simp [IsSelfAdjoint], ha.star.mul ha⟩ |
|
|
| theorem continuousOn_fugledeKadisonDeterminant (τ : TracialState A) : |
| ContinuousOn (fugledeKadisonDeterminant τ) {a : A | IsUnit a} := by |
| have hc : Continuous (fun b : A => |
| ENNReal.ofReal (Real.exp ((τ b).re / 2))) := by |
| apply ENNReal.continuous_ofReal.comp |
| fun_prop |
| have h := hc.comp_continuousOn (continuousOn_log_normSquare (A := A)) |
| apply h.congr |
| intro a ha |
| exact fugledeKadisonDeterminant_of_isUnit τ a ha |
|
|
| theorem continuousAt_fugledeKadisonDeterminant_of_isUnit |
| (τ : TracialState A) {a : A} (ha : IsUnit a) : |
| ContinuousAt (fugledeKadisonDeterminant τ) a := |
| (continuousOn_fugledeKadisonDeterminant τ a ha).continuousAt |
| (Units.isOpen.mem_nhds ha) |
|
|
| theorem continuous_fugledeKadisonDeterminant_one_sub_smul |
| (τ : TracialState A) {a : A} (ha : NormQuasinilpotent a) : |
| Continuous (fun z : ℂ => fugledeKadisonDeterminant τ (1 - z • a)) := by |
| rw [continuous_iff_continuousAt] |
| intro z |
| exact ContinuousAt.comp (f := fun w : ℂ => (1 : A) - w • a) |
| (continuousAt_fugledeKadisonDeterminant_of_isUnit τ (ha.isUnit_one_sub_smul z)) |
| (by fun_prop) |
|
|
| section Vector |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| theorem continuousOn_vectorFugledeKadisonDeterminant (ξ : H) : |
| ContinuousOn (vectorFugledeKadisonDeterminant ξ) |
| {T : H →L[ℂ] H | IsUnit T} := by |
| have hc : Continuous (fun B : H →L[ℂ] H => |
| ENNReal.ofReal (Real.exp ((vectorTrace ξ B).re / 2))) := by |
| apply ENNReal.continuous_ofReal.comp |
| unfold vectorTrace |
| fun_prop |
| have h := hc.comp_continuousOn (continuousOn_log_normSquare (A := H →L[ℂ] H)) |
| apply h.congr |
| intro T hT |
| exact vectorFugledeKadisonDeterminant_of_isUnit ξ T hT |
|
|
| theorem continuousAt_vectorFugledeKadisonDeterminant_of_isUnit |
| (ξ : H) {T : H →L[ℂ] H} (hT : IsUnit T) : |
| ContinuousAt (vectorFugledeKadisonDeterminant ξ) T := |
| (continuousOn_vectorFugledeKadisonDeterminant ξ T hT).continuousAt |
| (Units.isOpen.mem_nhds hT) |
|
|
| theorem continuous_vectorFugledeKadisonDeterminant_one_sub_smul |
| (ξ : H) {T : H →L[ℂ] H} (hT : NormQuasinilpotent T) : |
| Continuous (fun z : ℂ => vectorFugledeKadisonDeterminant ξ (1 - z • T)) := by |
| rw [continuous_iff_continuousAt] |
| intro z |
| exact ContinuousAt.comp (f := fun w : ℂ => (1 : H →L[ℂ] H) - w • T) |
| (continuousAt_vectorFugledeKadisonDeterminant_of_isUnit ξ (hT.isUnit_one_sub_smul z)) |
| (by fun_prop) |
|
|
| end Vector |
| end Invariant233 |
|
|
| end |
|
|
| end OAI |
|
|
