File size: 2,714 Bytes
f1f7953 | 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 64 65 66 67 | import OAI.Analysis.IrrationalRotation.Spectral.DeterminantNormBound
namespace OAI
noncomputable section
open Filter
open scoped ENNReal Topology CStarAlgebra
namespace Invariant233
variable {A : Type*} [CStarAlgebra A]
theorem regularizedDeterminant_monotone (τ : TracialState A) (a : A) :
Monotone (fun ε : {r : ℝ // 0 < r} => ENNReal.ofReal
(Real.exp ((τ (CFC.log (star a * a + ε.val • (1 : A)))).re / 2))) := by
let : PartialOrder A := CStarAlgebra.spectralOrder A
let : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
intro ε δ hεδ
apply ENNReal.ofReal_le_ofReal
apply Real.exp_le_exp.mpr
apply div_le_div_of_nonneg_right _ (by norm_num)
apply τ.real_mono
apply CFC.log_le_log
· apply add_le_add le_rfl
exact smul_le_smul_of_nonneg_right hεδ zero_le_one
· apply (StarOrderedRing.isStrictlyPositive_iff_spectrum_pos
(R := ℝ) _ (by simp [IsSelfAdjoint])).mpr
exact regularizedNormSquare_spectrum_pos a ε.val ε.property
theorem tendsto_positive_atBot_of_tendsto_zero {ι : Type*} {l : Filter ι}
(ε : ι → ℝ) (hε : ∀ i, 0 < ε i) (hlim : Tendsto ε l (𝓝 0)) :
Tendsto (fun i => (⟨ε i, hε i⟩ : {r : ℝ // 0 < r})) l atBot := by
apply tendsto_atBot.2
intro δ
filter_upwards [hlim.eventually (gt_mem_nhds δ.property)] with i hi
show ε i ≤ δ.val
exact hi.le
theorem tendsto_monotone_positive_iInf (f : {r : ℝ // 0 < r} → ℝ≥0∞)
(hf : Monotone f) {ι : Type*} {l : Filter ι}
(ε : ι → ℝ) (hε : ∀ i, 0 < ε i) (hlim : Tendsto ε l (𝓝 0)) :
Tendsto (fun i => f ⟨ε i, hε i⟩) l (𝓝 (⨅ δ, f δ)) := by
have h : Tendsto f atBot (𝓝 (⨅ δ, f δ)) := tendsto_atBot_iInf hf
exact h.comp (tendsto_positive_atBot_of_tendsto_zero ε hε hlim)
theorem tendsto_regularizedDeterminant
(τ : TracialState A) (a : A) {ι : Type*} {l : Filter ι}
(ε : ι → ℝ) (hε : ∀ i, 0 < ε i) (hlim : Tendsto ε l (𝓝 0)) :
Tendsto (fun i => ENNReal.ofReal
(Real.exp ((τ (CFC.log (star a * a + ε i • (1 : A)))).re / 2)))
l (𝓝 (fugledeKadisonDeterminant τ a)) := by
unfold fugledeKadisonDeterminant
exact tendsto_monotone_positive_iInf _ (regularizedDeterminant_monotone τ a) ε hε hlim
theorem tendsto_regularizedDeterminant_reciprocal (τ : TracialState A) (a : A) :
Tendsto (fun n : ℕ => ENNReal.ofReal
(Real.exp ((τ (CFC.log (star a * a + (1 / ((n : ℝ) + 1)) • (1 : A)))).re / 2)))
atTop (𝓝 (fugledeKadisonDeterminant τ a)) := by
exact tendsto_regularizedDeterminant τ a _ (fun n => by positivity)
(tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
end Invariant233
end
end OAI
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