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ddc84c5 | 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 | 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
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