| import Mathlib |
| import OAI.Analysis.BrownMeasure.Model |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter InnerProductSpace |
| open scoped ComplexInnerProductSpace Topology ENNReal Ring |
| namespace FiniteFactor |
| local instance unitDeterminantRealModule : Module ℝ ProductOp := Module.complexToReal ProductOp |
| local instance unitDeterminantFunctionalCalculus : |
| IsometricContinuousFunctionalCalculus ℝ ProductOp IsSelfAdjoint := |
| IsSelfAdjoint.instIsometricContinuousFunctionalCalculus |
|
|
| lemma sourceVacuumFunctional_mono {A B : ProductOp} (h : A≤B) : |
| (sourceVacuumFunctional A).re≤(sourceVacuumFunctional B).re := by |
| have hpos := (ContinuousLinearMap.nonneg_iff_isPositive (f := B-A)).mp |
| (sub_nonneg.mpr h) |
| have hb := hpos.re_inner_nonneg_right sourceVacuum |
| change 0≤(inner ℂ sourceVacuum ((B-A) sourceVacuum)).re at hb |
| simp only [sub_apply,inner_sub_right,Complex.sub_re] at hb |
| exact sub_nonneg.mp hb |
|
|
| lemma product_abs_isUnit {T : ProductOp} (hT : IsUnit T) : IsUnit (CFC.abs T) := by |
| apply (isUnit_pow_iff (by norm_num : (2:ℕ)≠0)).mp |
| rw [pow_two, CFC.abs_mul_abs] |
| exact hT.star.mul hT |
|
|
| lemma product_abs_spectrum_pos {T : ProductOp} (hT : IsUnit T) {y : ℝ} |
| (hy : y∈spectrum ℝ (CFC.abs T)) : 0<y := by |
| refine (spectrum_nonneg_of_nonneg (CFC.abs_nonneg T) hy).lt_of_ne ?_ |
| intro he |
| exact spectrum.zero_notMem ℝ (product_abs_isUnit hT) (he ▸ hy) |
|
|
| lemma logarithmCutoff_cfc_eq_log {A : ProductOp} (hA : 0≤A) (hu : IsUnit A) : |
| ∃ n:ℕ,cfc (logarithmCutoff n) A=CFC.log A := by |
| have hself := hA.isSelfAdjoint |
| obtain ⟨y,hy,hmin⟩ := (spectrum.isCompact (𝕜:=ℝ) A).exists_isMinOn |
| (ContinuousFunctionalCalculus.spectrum_nonempty A hself) continuous_id.continuousOn |
| have hy0 : 0<y := by |
| refine (spectrum_nonneg_of_nonneg hA hy).lt_of_ne ?_ |
| intro he |
| exact spectrum.zero_notMem ℝ hu (he ▸ hy) |
| have ht := Real.tendsto_exp_neg_atTop_nhds_zero.comp tendsto_natCast_atTop_atTop |
| obtain ⟨n,hn⟩ := (ht.eventually (gt_mem_nhds hy0)).exists |
| refine ⟨n,?_⟩ |
| apply cfc_congr |
| intro x hx |
| change Real.log (max (Real.exp (-(n:ℝ))) x)=Real.log x |
| have hn' : Real.exp (-(n:ℝ))≤x := hn.le.trans (hmin hx) |
| rw [max_eq_right hn'] |
|
|
| lemma FugledeKadisonDeterminant_of_isUnit (T : ProductOp) (hu : IsUnit T) : |
| FugledeKadisonDeterminant sourceVacuumFunctional T= |
| ENNReal.ofReal (Real.exp (sourceVacuumFunctional (CFC.log (CFC.abs T))).re) := by |
| have hp : ∀y∈spectrum ℝ (CFC.abs T),0<y := fun y hy=>product_abs_spectrum_pos hu hy |
| have hc : ContinuousOn Real.log (spectrum ℝ (CFC.abs T)) := by |
| apply Real.continuousOn_log.mono |
| intro y hy |
| exact (hp y hy).ne' |
| apply le_antisymm |
| · obtain ⟨n,hn⟩ := logarithmCutoff_cfc_eq_log (CFC.abs_nonneg T) (product_abs_isUnit hu) |
| exact (iInf_le _ n).trans_eq |
| (congrArg (fun operator : ProductOp => |
| ENNReal.ofReal (Real.exp (sourceVacuumFunctional operator).re)) hn) |
| · apply le_iInf |
| intro n |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| apply sourceVacuumFunctional_mono |
| apply cfc_mono (hf:=hc) (hg:=(logarithmCutoff n).continuous.continuousOn) |
| intro y hy |
| exact Real.log_le_log (hp y hy) (le_max_right _ _) |
| end FiniteFactor |
| end |
|
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| end OAI |
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