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import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.DimensionDrop.Limit
import OAI.Analysis.JiangSu.DimensionDrop.TraceExistence
namespace OAI
noncomputable section
namespace SequentialCStar
open Pureness Filter
open scoped Topology
variable {D : Type*} [CStarAlgebra D]
lemma selfAdjoint_span_star {g x : D} (hg : IsSelfAdjoint g)
(hx : x∈TwoSidedIdeal.span {g}) : star x∈TwoSidedIdeal.span {g} := by
induction hx using TwoSidedIdeal.span_induction with
| mem x hx =>
rw [Set.mem_singleton_iff.mp hx,hg.star_eq]
exact TwoSidedIdeal.subset_span (Set.mem_singleton g)
| zero => simp
| add x y _ _ hx hy => simpa only [star_add] using (TwoSidedIdeal.span {g}).add_mem hx hy
| neg x _ hx => simpa only [star_neg] using (TwoSidedIdeal.span {g}).neg_mem hx
| left_absorb a x _ hx =>
simpa only [star_mul] using (TwoSidedIdeal.span {g}).mul_mem_right (star x) (star a) hx
| right_absorb a x _ hx =>
simpa only [star_mul] using (TwoSidedIdeal.span {g}).mul_mem_left (star a) (star x) hx
def generatedClosedStarIdeal {g : D} (hg : IsSelfAdjoint g) : ClosedStarIdeal D where
ideal := closedTwoSidedIdeal (TwoSidedIdeal.span {g})
isClosed := by rw [coe_closedTwoSidedIdeal]; exact isClosed_closure
star_mem := by
intro x hx
rw [←SetLike.mem_coe,coe_closedTwoSidedIdeal] at hx ⊢
obtain ⟨u,hu,hux⟩ := mem_closure_iff_seq_limit.mp hx
exact isClosed_closure.mem_of_tendsto hux.star
(Eventually.of_forall (fun n => subset_closure (selfAdjoint_span_star hg (hu n))))
lemma generatedClosedStarIdeal_mem {g : D} (hg : IsSelfAdjoint g) :
g∈(generatedClosedStarIdeal hg).ideal := by
change g∈closedTwoSidedIdeal (TwoSidedIdeal.span {g})
rw [←SetLike.mem_coe,coe_closedTwoSidedIdeal]
exact subset_closure (TwoSidedIdeal.subset_span (Set.mem_singleton g))
lemma generatedClosedStarIdeal_le {g : D} (hg : IsSelfAdjoint g)
(I : TwoSidedIdeal D) (hI : IsClosed (I:Set D)) (hgI : g∈I) :
(generatedClosedStarIdeal hg).ideal≤I := by
change closedTwoSidedIdeal (TwoSidedIdeal.span {g})≤I
rw [TwoSidedIdeal.le_iff,coe_closedTwoSidedIdeal]
apply closure_minimal _ hI
exact TwoSidedIdeal.span_le.mpr (by simpa)
namespace System
variable {C : ℕ → Type*} [∀ n,CStarAlgebra (C n)] [∀ n,Nontrivial (C n)]
variable (S : System C)
def EventuallyFull : Prop := ∀ n (a : C n),a≠0 →
∃ (m : ℕ) (h : n≤ m),∀ I : TwoSidedIdeal (C m),S.map h a∈I → I=⊤
variable [PartialOrder S.Limit] [StarOrderedRing S.Limit]
lemma closedStarIdeal_trivial (hS : S.EventuallyFull) (J : ClosedStarIdeal S.Limit) :
J.ideal=⊥ ∨ J.ideal=⊤ := by
by_cases hJ : J.ideal=⊤
· exact Or.inr hJ
left
let π : S.Limit →⋆ₐ[ℂ] J.Quotient :=
{ J.quotientMap with map_star' := J.quotientMap_star }
have hker (x : S.Limit) : π x=0 ↔ x∈J.ideal := by
change J.ideal.ringCon.mk' x=J.ideal.ringCon.mk' 0 ↔ x∈J.ideal
exact ⟨Quotient.exact,Quotient.sound⟩
have hπ1 : π 1≠0 := by
intro h
exact hJ (J.ideal.eq_top ((hker 1).mp h))
have hinj (n : ℕ) : Function.Injective (π.comp (S.inclusion n)) := by
apply (π.comp (S.inclusion n)).toLinearMap.ker_eq_bot.mp
apply le_antisymm _ bot_le
intro a ha
change a=0
by_contra ha0
obtain ⟨m,hnm,hfull⟩ := hS n a ha0
let I := TwoSidedIdeal.ker (π.comp (S.inclusion m))
have he : S.map hnm a∈I := by
apply (TwoSidedIdeal.mem_ker _).mpr
change π (S.inclusion m (S.map hnm a))=0
rw [S.inclusion_compatible]
exact ha
have hI := hfull I he
have h1 : (1:C m)∈I := I.one_mem hI
apply hπ1
have he1 := (TwoSidedIdeal.mem_ker _).mp h1
change π (S.inclusion m 1)=0 at he1
simpa using he1
have hnorm (n : ℕ) (a : C n) : ‖π (S.inclusion n a)‖=‖a‖ :=
NonUnitalStarAlgHom.norm_map (π.comp (S.inclusion n)).toNonUnitalStarAlgHom (hinj n) a
apply le_antisymm _ bot_le
intro x hx
change x=0
apply norm_eq_zero.mp
apply le_antisymm _ (norm_nonneg _)
apply le_of_forall_pos_le_add
intro ε hε
obtain ⟨n,a,ha⟩ := S.finite_stage_approximation x (half_pos hε)
have hπx : π x=0 := (hker x).mpr hx
have hA : ‖a‖<ε/2 := by
rw [←hnorm n a]
have h := NonUnitalStarAlgHom.norm_apply_le π (S.inclusion n a-x)
rw [map_sub,hπx,sub_zero,norm_sub_rev] at h
exact h.trans_lt ha
have h := norm_add_le (x-S.inclusion n a) (S.inclusion n a)
rw [sub_add_cancel,S.inclusion_norm] at h
linarith
theorem limit_topologicallySimple (hS : S.EventuallyFull) : TopologicallySimple S.Limit := by
intro I hI
by_cases h0 : I=⊥
· exact Or.inl h0
right
obtain ⟨x,hx,hx0⟩ := SetLike.exists_of_lt (bot_lt_iff_ne_bot.mpr h0)
have hxne : x≠0 := by simpa using hx0
have hg : IsSelfAdjoint (star x*x) := IsSelfAdjoint.star_mul_self x
let J := generatedClosedStarIdeal hg
have hJI : J.ideal≤I := generatedClosedStarIdeal_le hg I hI (I.mul_mem_left _ _ hx)
rcases S.closedStarIdeal_trivial hS J with hbot|htop
· have hz := generatedClosedStarIdeal_mem hg
change star x*x∈J.ideal at hz
rw [hbot,TwoSidedIdeal.mem_bot] at hz
have hn := CStarRing.norm_star_mul_self (x := x)
rw [hz,norm_zero] at hn
have : ‖x‖=0 := by nlinarith [norm_nonneg x]
exact False.elim (hxne (norm_eq_zero.mp this))
· exact top_unique (htop ▸ hJI)
end System
end SequentialCStar
end
noncomputable section
namespace OriginalJiangSu
open JiangSuConstruction
open scoped ComplexOrder CStarAlgebra
lemma system_eventuallyFull : system.EventuallyFull := by
intro n f hf
obtain ⟨m,h,hf'⟩ := eventually_fiberwise_nonzero n f hf
refine ⟨m,h,?_⟩
intro I hI
exact dimensionDrop_full_of_fiberwise_nonzero _ hf' I hI
theorem simple : Pureness.TopologicallySimple Algebra :=
system.limit_topologicallySimple system_eventuallyFull
end OriginalJiangSu
end
end OAI
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