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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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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