| import OAI.Analysis.Hyperinvariant.ProjectionBridge |
| import OAI.Analysis.FiniteFactor.CrossedProduct.PreparedFactor |
| import OAI.Analysis.FiniteFactor.CrossedProduct.InvariantProjection |
| import OAI.Analysis.FiniteFactor.CrossedProduct.HilbertSeparable |
| import OAI.Analysis.FiniteFactor.Matrix.Entries |
|
|
| namespace OAI |
|
|
| namespace Hyperinvariant |
| open FiniteFactor MeasureTheory Filter |
| open scoped Topology |
| noncomputable section |
|
|
| / |
| theorem source_basis_orthonormal : Orthonormal ℂ sourceBasis := by |
| classical |
| apply orthonormal_iff_ite.mpr |
| intro i j |
| rw [sourceBasis, sourceBasis, sourceFiberIndicator, sourceFiberIndicator, |
| L2.inner_indicatorConstLp_one_indicatorConstLp_one |
| (hμs := sourceFiberMeasure_ne_top Set.univ i) |
| (hμt := sourceFiberMeasure_ne_top Set.univ j)] |
| by_cases hij : i = j |
| · subst j |
| simp [measureReal_def, sourceMeasure, Measure.prod_prod] |
| · have he : ((Set.univ : Set (SourceCoordinate → ℝ)) ×ˢ {i}) ∩ |
| (Set.univ ×ˢ {j}) = ∅ := by |
| ext z |
| simp only [Set.mem_inter_iff, Set.mem_prod, Set.mem_univ, Set.mem_singleton_iff, |
| true_and, Set.mem_empty_iff_false, iff_false, not_and] |
| exact fun hi hj => hij (hi.symm.trans hj) |
| simp [he, hij] |
|
|
| theorem source_infiniteDimensional : ¬FiniteDimensional ℂ (SourceHilbert SourceCoordinate) := by |
| intro hfin |
| let : FiniteDimensional ℂ (SourceHilbert SourceCoordinate) := hfin |
| exact Module.Finite.not_linearIndependent_of_infinite sourceBasis |
| source_basis_orthonormal.linearIndependent |
|
|
| / |
| whose ordinary commutant has no nontrivial closed invariant subspace. -/ |
| theorem source_mainClaim : MainClaim (H := SourceHilbert SourceCoordinate) := by |
| obtain ⟨θ, hr, hphase, hs, _he, _hcenter, _hfinite, _hminimal, _hpredual, |
| hθ, hT, hT0, _hinj, _hcontract, hqn⟩ := exists_source_prepared_factor |
| refine ⟨sourceOperator θ sourceLogWeight hθ.nonneg sourceLogWeight_nonneg, |
| hT0, hqn, ?_⟩ |
| apply transitiveCommutant_of_invariantProjections (sourceAlgebra θ) _ hT |
| intro p hp hproj hinv |
| exact sourceInvariantProjection_trivial θ hθ hr hs hphase hp hproj hinv |
|
|
| end |
| end Hyperinvariant |
|
|
| end OAI |
|
|
