| import OAI.Analysis.FiniteFactor.Diffuse.SupportLimit |
| import OAI.Analysis.FiniteFactor.Diffuse.TracePowers |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology BigOperators |
| namespace Nonhyperlinear.Diffuseness |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| def vectorFunctional (v:H) : (H→L[ℂ]H)→ₗ[ℂ]ℂ where |
| toFun T := inner ℂ v (T v) |
| map_add' T S := inner_add_right v (T v) (S v) |
| map_smul' c T := inner_smul_right v (T v) c |
|
|
| theorem support_trace_eq (M:VonNeumannAlgebra H) (v:H) |
| (hcyclic:∀a b:M.toStarSubalgebra, |
| inner ℂ v (((a*b:M.toStarSubalgebra):H→L[ℂ]H) v)= |
| inner ℂ v (((b*a:M.toStarSubalgebra):H→L[ℂ]H) v)) |
| (T:M.toStarSubalgebra) (hT:‖(T:H→L[ℂ]H)‖≤1) : |
| inner ℂ v (rangeSupport (T:H→L[ℂ]H) v)= |
| inner ℂ v (rangeSupport (star (T:H→L[ℂ]H)) v) := by |
| let t:H→L[ℂ]H := T |
| let f:M.toStarSubalgebra→ₗ[ℂ]ℂ := |
| (vectorFunctional v).comp M.toStarSubalgebra.toSubmodule.subtype |
| have hf : ∀a b,f (a*b)=f (b*a) := hcyclic |
| have he : ∀n, inner ℂ v (cesaro (1-t*star t) n v)= |
| inner ℂ v (cesaro (1-star t*t) n v) := by |
| intro n |
| change vectorFunctional v (cesaro (1-t*star t) n)= |
| vectorFunctional v (cesaro (1-star t*t) n) |
| simp only [cesaro,map_smul,map_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro k hk |
| exact trace_defect_pow f hf T (star T) k |
| have hts : ‖star t‖≤1 := by simpa [t,ContinuousLinearMap.star_eq_adjoint] using hT |
| have hl := Filter.Tendsto.inner (𝕜:=ℂ) (tendsto_const_nhds (x:=v)) (cesaro_star_mul_tendsto (star t) hts v) |
| have hr := Filter.Tendsto.inner (𝕜:=ℂ) (tendsto_const_nhds (x:=v)) (cesaro_star_mul_tendsto t hT v) |
| have hlim : inner ℂ v ((star t).ker.starProjection v)= |
| inner ℂ v (t.ker.starProjection v) := by |
| apply tendsto_nhds_unique hl |
| apply hr.congr' |
| exact Filter.Eventually.of_forall (fun n=>by simpa only [star_star] using (he n).symm) |
| change inner ℂ v (rangeSupport t v)=inner ℂ v (rangeSupport (star t) v) |
| rw [rangeSupport_eq_one_sub_kernel,rangeSupport_eq_one_sub_kernel,star_star] |
| change inner ℂ v (v-(star t).ker.starProjection v)=inner ℂ v (v-t.ker.starProjection v) |
| rw [inner_sub_right,inner_sub_right,hlim] |
|
|
| end Nonhyperlinear.Diffuseness |
|
|
| end |
|
|
| end OAI |
|
|
