| import Mathlib |
| import OAI.Analysis.BrownMeasure.Model |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology ENNReal |
| namespace FiniteFactor |
| lemma spectrum_eq_zero_of_root_norm_tendsto {A : Type*} [NormedRing A] [NormedAlgebra ℂ A] |
| [CompleteSpace A] [Nontrivial A] (T : A) |
| (hT : Tendsto (fun n:ℕ=>‖T^n‖^(1/(n:ℝ))) atTop (𝓝 0)) : spectrum ℂ T={0} := by |
| have hr : spectralRadius ℂ T=0 := by |
| apply tendsto_nhds_unique (spectrum.pow_norm_pow_one_div_tendsto_nhds_spectralRadius T) |
| simpa only [ENNReal.ofReal_zero,Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp hT |
| have hsub : spectrum ℂ T⊆{0} := by |
| intro z hz |
| have hle : (‖z‖₊:ℝ≥0∞)≤ spectralRadius ℂ T := by |
| rw [spectralRadius_eq_of_unital] |
| exact le_iSup_of_le z (le_iSup_of_le hz le_rfl) |
| rw [hr] at hle |
| have hz0 : ‖z‖₊=0 := by exact_mod_cast (le_zero_iff.mp hle) |
| simpa only [Set.mem_singleton_iff,nnnorm_eq_zero] using hz0 |
| apply Set.Subset.antisymm hsub |
| rw [Set.singleton_subset_iff] |
| obtain ⟨z,hz⟩ := spectrum.nonempty T |
| have hz0 : z=0 := hsub hz |
| simpa only [hz0] using hz |
| end FiniteFactor |
| end |
|
|
| end OAI |
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