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9eeb9c7 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Analysis.IrrationalRotation.Operators.HyperinvariantProjection
import OAI.Analysis.IrrationalRotation.Analysis.UnitaryTransport
import OAI.Analysis.IrrationalRotation.Operators.CommutantAlgebra
import OAI.Analysis.IrrationalRotation.Spectral.QuasinilpotentNonscalar
import OAI.Analysis.IrrationalRotation.Model.Analytic
import OAI.Analysis.IrrationalRotation.Operators.HilbertModelSize
import Mathlib.MeasureTheory.Function.L2Space
namespace OAI
noncomputable section
namespace Invariant233
open scoped Topology
open TopologicalSpace
variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
[CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace ℂ F]
[CompleteSpace F]
theorem no_hyperinvariant_subspace_of_no_invariant_projection
(M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M)
(hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p →
(1 - p) * T * p = 0 → p = 0 ∨ p = 1) :
∀ S : Submodule ℂ E, IsClosed (S : Set E) →
(∀ A : E →L[ℂ] E, Commute A T → ∀ x ∈ S, A x ∈ S) → S = ⊥ ∨ S = ⊤ := by
intro S hS hhyper
let : CompleteSpace S := hS.completeSpace_coe
have hhyper' : ∀ A : E →L[ℂ] E, Commute A T → S ∈ Module.End.invtSubmodule A :=
hhyper
obtain ⟨hmem, hinv⟩ := hyperinvariant_subspace_projection_mem M hT S hhyper'
rcases hproj S.starProjection hmem isStarProjection_starProjection hinv with hzero | hone
· left
apply le_antisymm _ bot_le
intro x hx
have heq := Submodule.starProjection_eq_self_iff.mpr hx
simpa [hzero] using heq.symm
· right
apply le_antisymm le_top
intro x _
apply Submodule.starProjection_eq_self_iff.mp
simp [hone]
theorem hilbert_counterexample_of_factor_witness
[SeparableSpace E] [SeparableSpace F]
(hE : ¬ FiniteDimensional ℂ E) (hF : ¬ FiniteDimensional ℂ F)
(M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M) (hne : T ≠ 0)
(hquasi : Filter.Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ)))
Filter.atTop (𝓝 (0 : ℝ)))
(hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p →
(1 - p) * T * p = 0 → p = 0 ∨ p = 1) :
∃ T' : F →L[ℂ] F, T' ≠ 0 ∧
Filter.Tendsto (fun n : ℕ => ‖T' ^ n‖ ^ (1 / (n : ℝ)))
Filter.atTop (𝓝 (0 : ℝ)) ∧
(∀ S : Submodule ℂ F, IsClosed (S : Set F) →
(∀ A : F →L[ℂ] F, Commute A T' → ∀ x ∈ S, A x ∈ S) → S = ⊥ ∨ S = ⊤) := by
obtain ⟨e⟩ := UnitaryTransport.nonempty_linearIsometryEquiv hE hF
exact UnitaryTransport.transport_counterexample e T hne hquasi
(no_hyperinvariant_subspace_of_no_invariant_projection M hT hproj)
theorem hasNoNontrivialHyperinvariantSubspace_of_no_invariant_projection
(M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M)
(hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p →
(1 - p) * T * p = 0 → p = 0 ∨ p = 1) :
HasNoNontrivialHyperinvariantSubspace T := by
intro S hS
apply no_hyperinvariant_subspace_of_no_invariant_projection M hT hproj S hS.1
intro A hA
apply hS.2 A
exact (operatorCommutant_mem_iff T A).2 hA.symm.eq
theorem transitive_commutant_of_factor_witness
[SeparableSpace E] [SeparableSpace F]
(hE : ¬ FiniteDimensional ℂ E) (hF : ¬ FiniteDimensional ℂ F)
(M : VonNeumannAlgebra E) {T : E →L[ℂ] E} (hT : T ∈ M) (hne : T ≠ 0)
(hquasi : NormQuasinilpotent T)
(hproj : ∀ p : E →L[ℂ] E, p ∈ M → IsStarProjection p →
(1 - p) * T * p = 0 → p = 0 ∨ p = 1) :
∃ T' : F →L[ℂ] F, T' ≠ 0 ∧ NormQuasinilpotent T' ∧
HasNoNontrivialHyperinvariantSubspace T' ∧
operatorCommutant T' ≠ ⊤ ∧ IsStronglyClosed (operatorCommutant T') ∧
IsTransitive (operatorCommutant T') := by
classical
have : Nontrivial F := by
cases subsingleton_or_nontrivial F with
| inl h =>
let : Subsingleton F := h
exact False.elim (hF inferInstance)
| inr h => exact h
obtain ⟨T', hne', hquasi', hhyper'⟩ :=
hilbert_counterexample_of_factor_witness hE hF M hT hne hquasi hproj
have hnonscalar : ¬ ∃ c : ℂ, T' = c • (1 : F →L[ℂ] F) := by
rintro ⟨c, hc⟩
exact InvariantProjection.continuousLinearMap_not_scalar_of_nonzero_norm_root_tendsto
hne' hquasi' c hc
have hhyper : HasNoNontrivialHyperinvariantSubspace T' := by
intro S hS
apply hhyper' S hS.1
intro A hA
exact hS.2 A ((operatorCommutant_mem_iff T' A).2 hA.symm.eq)
obtain ⟨hproper, hclosed, htrans⟩ := commutant_algebra_properties T' hnonscalar
exact ⟨T', hne', hquasi', hhyper, hproper, hclosed, htrans.2 hhyper⟩
theorem transitive_commutant_of_circle_witness
[SeparableSpace F] (hF : ¬ FiniteDimensional ℂ F)
(M : VonNeumannAlgebra (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure))
{T : (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure) →L[ℂ]
(MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure)}
(hT : T ∈ M) (hne : T ≠ 0) (hquasi : NormQuasinilpotent T)
(hproj : ∀ p : (MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure) →L[ℂ]
(MeasureTheory.Lp (lp (fun _ : ℤ => ℂ) 2) 2 circleMeasure),
p ∈ M → IsStarProjection p → (1 - p) * T * p = 0 → p = 0 ∨ p = 1) :
∃ T' : F →L[ℂ] F, T' ≠ 0 ∧ NormQuasinilpotent T' ∧
HasNoNontrivialHyperinvariantSubspace T' ∧
operatorCommutant T' ≠ ⊤ ∧ IsStronglyClosed (operatorCommutant T') ∧
IsTransitive (operatorCommutant T') := by
let := HilbertModelSize.circle_representation_separable
exact transitive_commutant_of_factor_witness
HilbertModelSize.circle_representation_not_finiteDimensional hF M hT hne hquasi hproj
end Invariant233
end
end OAI
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