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110f4f3 | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import Mathlib
import OAI.Analysis.BoundedHochschild.CocycleNormalParts
import OAI.Analysis.BoundedHochschild.FactorFinitePaths
import OAI.Analysis.BoundedHochschild.FreeWords
import OAI.Analysis.BoundedHochschild.MatrixDisplacement
import OAI.Analysis.BoundedHochschild.UnitaryLifts
import OAI.Analysis.BoundedHochschild.FactorPrimitive
import OAI.Analysis.BoundedHochschild.TypeIICorners
import OAI.Analysis.BoundedHochschild.CenterPowers
import OAI.Analysis.BoundedHochschild.CentralAtoms
namespace OAI
/-! Orthogonal atomic central resolution. -/
noncomputable section
open Filter Topology
open scoped ComplexOrder
namespace HochschildAtomic
open BoundedHochschild HochschildTypeII FinitePathMain
universe u
variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M]
[NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X]
def AtomicCenter (M : Type u) [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] : Prop :=
∀ p : M, IsStarProjection p → (∀ a : M, Commute p a) → p ≠ 0 →
∃ q : M, CentralAtom q ∧ q ≤ p
lemma central_atoms_orthogonal {p q : M} (hp : CentralAtom p) (hq : CentralAtom q) (hne : p ≠ q) :
p*q=0 := by
have hpq : IsStarProjection (p*q) := hp.1.mul hq.1 (hp.2.1 q)
have hc : ∀ b : M, Commute (p*q) b := fun b => (hp.2.1 b).mul_left (hq.2.1 b)
have hpqp : p*q ≤ p := (hpq.le_iff_mul_eq_right hp.1).mpr
(by rw [← mul_assoc,hp.1.isIdempotentElem.eq])
have hpqq : p*q ≤ q := (hpq.le_iff_mul_eq_left hq.1).mpr
(by rw [mul_assoc,hq.1.isIdempotentElem.eq])
rcases hp.2.2.2 (p*q) hpq hc hpqp with hz | hpp
· exact hz
rcases hq.2.2.2 (p*q) hpq hc hpqq with hz | hqq
· exact hz
exact (hne (hpp.symm.trans hqq)).elim
variable (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) (τ : M →ₚ[ℂ] ℂ)
include e in
theorem central_atoms_total
(hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a))
(hfaith : ∀ a : M, τ (star a*a)=0 → a=0) (hatomic : AtomicCenter M)
(x : M) (hx : ∀ p : M, CentralAtom p → p*x=0) : x=0 := by
classical
let S : Set M := {p | CentralProjection p ∧ p*x=0}
have hzero : (0:M) ∈ S := ⟨⟨IsStarProjection.zero M,fun _ => Commute.zero_left _⟩,zero_mul _⟩
obtain ⟨q,_hq0,hmax⟩ := zorn_le_nonempty₀ S (fun c hc hchain y hy => by
obtain ⟨q,hq,hqx,hupper⟩ := central_annihilator_chain e τ hτ htr hfaith
x c hchain ⟨y,hy⟩ (fun p hp => hc hp)
exact ⟨q,⟨hq,hqx⟩,hupper⟩) 0 hzero
have hq : CentralProjection q := hmax.1.1
have hqx : q*x=0 := hmax.1.2
have hone : q=1 := by
by_contra hne
have hr : CentralProjection (1-q) :=
⟨hq.1.one_sub,fun a => (Commute.one_left a).sub_left (hq.2 a)⟩
obtain ⟨p,hp,hpq⟩ := hatomic (1-q) hr.1 hr.2 (sub_ne_zero.mpr (Ne.symm hne))
have hqp : q*p=0 := projection_orthogonal_of_le_complement hq.1 hp.1 hpq
have hnew : q+p ∈ S := ⟨⟨hq.1.add hp.1 hqp,fun a => (hq.2 a).add_left (hp.2.1 a)⟩,
by rw [add_mul,hqx,hx p hp,zero_add]⟩
have he : q+p=q := le_antisymm (hmax.2 hnew (le_add_of_nonneg_right hp.1.nonneg))
(le_add_of_nonneg_right hp.1.nonneg)
exact hp.2.2.1 (add_eq_left.mp he)
simpa only [hone,one_mul] using hqx
include e in
theorem typeII_no_minimal
(hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a))
(hfaith : ∀ a : M, τ (star a*a)=0 → a=0) (hII : TypeII1 M) :
HochschildDiffuse.NoMinimalProjections M := by
intro p hp hp0
by_contra hn
have hmin (q : M) (hq : IsStarProjection q) (hqp : q ≤ p) : q=0 ∨ q=p := by
by_cases hq0 : q=0
· exact Or.inl hq0
· right
by_contra hne
exact hn ⟨q,hq,hqp,hq0,hne⟩
have hab : AbelianProjection p := by
refine ⟨hp,fun a b => ?_⟩
apply HochschildCorners.commute_corner_of_projections e hp
(b:=p*b*p) _ (HochschildCorners.compress_mem_corner hp a)
intro q hq hqp
rcases hmin q hq hqp with hz | he
· rw [hz]; exact Commute.zero_left _
· rw [he]
change p*(p*b*p)=(p*b*p)*p
rw [(HochschildCorners.compress_mem_corner hp b).1,
(HochschildCorners.compress_mem_corner hp b).2]
exact hp0 (no_abelian_projection e τ hτ htr hfaith hII hab)
end HochschildAtomic
end
end OAI
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