| import OAI.Analysis.Hyperinvariant.Specification | |
| namespace OAI | |
| namespace Hyperinvariant | |
| universe u | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| /-- Triviality of invariant projections in a von Neumann algebra implies | |
| hyperinvariant-subspace triviality in the entire bounded-operator algebra. -/ | |
| theorem transitiveCommutant_of_invariantProjections | |
| (M : VonNeumannAlgebra H) (T : H →L[ℂ] H) (hT : T ∈ M) | |
| (hproj : ∀ p ∈ M, IsStarProjection p → (1 - p) * T * p = 0 → | |
| p = 0 ∨ p = 1) : TransitiveCommutant T := by | |
| intro K hK hInv | |
| let : CompleteSpace K := hK.completeSpace_coe | |
| have hp : IsStarProjection K.starProjection := isStarProjection_starProjection | |
| have hpM : K.starProjection ∈ M := by | |
| apply (VonNeumannAlgebra.IsStarProjection.mem_iff hp M).2 | |
| intro A hA | |
| rw [Submodule.range_starProjection] | |
| rw [Module.End.mem_invtSubmodule_iff_forall_mem_of_mem] | |
| exact hInv A ((VonNeumannAlgebra.mem_commutant_iff.mp hA) T hT).symm | |
| have hi : (1 - K.starProjection) * T * K.starProjection = 0 := by | |
| ext x | |
| have hm : T (K.starProjection x) ∈ K := | |
| hInv T rfl _ (K.starProjection_apply_mem x) | |
| simp only [mul_apply_eq_comp, sub_apply, one_apply_eq_self, zero_apply] | |
| exact sub_eq_zero.mpr (Submodule.starProjection_eq_self_iff.mpr hm).symm | |
| rcases hproj K.starProjection hpM hp hi with hz | ho | |
| · left | |
| apply le_antisymm _ bot_le | |
| intro x hx | |
| have hf : K.starProjection x = x := Submodule.starProjection_eq_self_iff.mpr hx | |
| have he : K.starProjection x = 0 := congrArg (fun A : H →L[ℂ] H => A x) hz | |
| exact hf.symm.trans he | |
| · right | |
| apply le_antisymm le_top | |
| intro x _ | |
| have he : K.starProjection x = x := congrArg (fun A : H →L[ℂ] H => A x) ho | |
| have hm := K.starProjection_apply_mem x | |
| rwa [he] at hm | |
| end Hyperinvariant | |
| end OAI | |