| import OAI.Analysis.FiniteFactor.Diffuse.SupportTrace |
| import OAI.Analysis.FiniteFactor.Diffuse.ProjectionTrace |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace Nonhyperlinear.Diffuseness |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| /-- A faithful tracial concrete algebra with finite identity partitions of |
| arbitrarily small projection trace has no minimal projection. -/ |
| theorem no_minimal_projection_of_small_partitions (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)) |
| (hfaithful:∀T:M.toStarSubalgebra,(T:H→L[ℂ]H) v=0→T=0) |
| (hpart:∀ε:ℝ,0<ε→∃n:ℕ,∃e:Fin n→M.toStarSubalgebra, |
| (∀j,IsStarProjection (e j:H→L[ℂ]H)) ∧ (∑j,e j)=1 ∧ |
| ∀j,(inner ℂ v ((e j:H→L[ℂ]H) v)).re<ε) |
| (p:M.toStarSubalgebra) (hps:star p=p) (hpi:p*p=p) (hp0:p≠0) : |
| ∃q:M.toStarSubalgebra,star q=q ∧ q*q=q ∧ q≠0 ∧ q≠p ∧ q*p=q := by |
| classical |
| have hpp:IsStarProjection (p:H→L[ℂ]H) := |
| isStarProjection_iff'.mpr ⟨congrArg Subtype.val hpi,congrArg Subtype.val hps⟩ |
| have hpv:(p:H→L[ℂ]H) v≠0 := fun hz=>hp0 (hfaithful p hz) |
| have htpos:0<(inner ℂ v ((p:H→L[ℂ]H) v)).re := by |
| rw [projection_vector_trace v hpp] |
| simp only [←Complex.ofReal_pow,Complex.ofReal_re] |
| exact sq_pos_of_pos (norm_pos_iff.mpr hpv) |
| obtain ⟨n,e,hep,hes,het⟩:=hpart _ htpos |
| obtain ⟨j,hj⟩:∃j,p*e j≠0 := by |
| by_contra hn |
| have hh:∀j,p*e j=0 := by simpa only [not_exists,not_not] using hn |
| apply hp0 |
| calc |
| p=p*1 := (mul_one p).symm |
| _=p*(∑j,e j) := by rw [hes] |
| _=∑j,p*e j := Finset.mul_sum _ _ _ |
| _=0 := by simp [hh] |
| by_contra hnone |
| let T:H→L[ℂ]H := (p*e j:M.toStarSubalgebra) |
| have hT0:T≠0 := fun hh=>hj (Subtype.ext hh) |
| have hTnorm:‖T‖≤1 := by |
| exact (norm_mul_le (p:H→L[ℂ]H) (e j:H→L[ℂ]H)).trans |
| ((mul_le_mul (IsStarProjection.norm_le _ hpp) (IsStarProjection.norm_le _ (hep j)) |
| (norm_nonneg _) zero_le_one).trans_eq (one_mul 1)) |
| have hpT:(p:H→L[ℂ]H)*T=T := by |
| change ((p*p)*e j:M.toStarSubalgebra).val=(p*e j).val |
| rw [hpi] |
| let q:M.toStarSubalgebra := ⟨rangeSupport T,rangeSupport_mem M (p*e j).property⟩ |
| have hqp:q*p=q := Subtype.ext (rangeSupport_mul_projection hpp hpT) |
| have hq0:q≠0 := fun hh=>rangeSupport_ne_zero hT0 (congrArg Subtype.val hh) |
| have hqeq:q=p := by |
| by_contra hh |
| exact hnone ⟨q,Subtype.ext (rangeSupport_projection T).isSelfAdjoint.star_eq, |
| Subtype.ext (rangeSupport_projection T).isIdempotentElem.eq,hq0,hh,hqp⟩ |
| have heT:(e j:H→L[ℂ]H)*star T=star T := by |
| change (e j:H→L[ℂ]H)*star ((p:H→L[ℂ]H)*(e j:H→L[ℂ]H))=star ((p:H→L[ℂ]H)*(e j:H→L[ℂ]H)) |
| rw [star_mul,hpp.isSelfAdjoint.star_eq,(hep j).isSelfAdjoint.star_eq,←mul_assoc, |
| (hep j).isIdempotentElem.eq] |
| have hright:=rangeSupport_mul_projection (hep j) heT |
| have hle:=projection_vector_trace_mono v (hep j) (rangeSupport_projection (star T)) hright |
| have heq:=support_trace_eq M v hcyclic (p*e j) hTnorm |
| have hleft:rangeSupport T=(p:H→L[ℂ]H) := congrArg Subtype.val hqeq |
| change inner ℂ v (rangeSupport T v)=inner ℂ v (rangeSupport (star T) v) at heq |
| rw [hleft] at heq |
| rw [←heq] at hle |
| exact (not_lt_of_ge hle) (het j) |
|
|
| end Nonhyperlinear.Diffuseness |
|
|
| end |
|
|
| end OAI |
|
|
