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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 | import OAI.Analysis.IrrationalRotation.Operators.NormalAffiliatedTransport
namespace OAI
noncomputable section
namespace Invariant233
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
[CompleteSpace E]
theorem IsAffiliated.inverse
{M : VonNeumannAlgebra E} {H : E →ₗ.[ℂ] E} (hH : IsAffiliated M H)
(hinj : Function.Injective H) : IsAffiliated M H.inverse := by
have hker : H.toFun.ker = ⊥ := LinearMap.ker_eq_bot.mpr hinj
have hpartialker : H.ker = ⊥ := by simp [LinearPMap.ker, hker]
intro U hU
have hgraph : ∀ y : H.inverse.domain,
((U : E →L[ℂ] E) (y : E), (U : E →L[ℂ] E) (H.inverse y)) ∈
H.inverse.graph := by
intro y
have hy : (y : E) ∈ H.toFun.range := by
simpa only [LinearPMap.inverse_domain] using y.property
obtain ⟨x, hx⟩ := hy
have hxy : H x = (y : E) := hx
have hyx : H.inverse y = (x : E) := LinearPMap.inverse_apply_eq hpartialker hxy
obtain ⟨hD, hC⟩ := hH.commutes_of_mem_commutant hU
rw [hyx, ← hxy, ← hC x]
exact LinearPMap.mem_inverse_graph hpartialker ⟨(U : E →L[ℂ] E) x, hD x⟩
let hD : ∀ y : H.inverse.domain, (U : E →L[ℂ] E) (y : E) ∈ H.inverse.domain :=
fun y => LinearPMap.mem_domain_of_mem_graph (hgraph y)
refine ⟨hD, fun y => ?_⟩
exact H.inverse.mem_graph_snd_inj (H.inverse.mem_graph ⟨(U : E →L[ℂ] E) y, hD y⟩)
(hgraph y) rfl
end Invariant233
end
end OAI
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