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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 | 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
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