| 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 | |