| import Mathlib | |
| import OAI.Analysis.Commutator.BicommutantRange | |
| import OAI.Analysis.Commutator.BiprodLinkColumn | |
| namespace OAI | |
| noncomputable section | |
| noncomputable section | |
| namespace Kadison.BiprodLinkColumn | |
| open HilbertBiprod PartialIsometryOrder | |
| variable {H G : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| [NormedAddCommGroup G] [InnerProductSpace ℂ G] [CompleteSpace G] | |
| variable {A : Type*} [CStarAlgebra A] | |
| lemma component_diagonal_intertwines | |
| (φ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (ψ : A →⋆ₐ[ℂ] (G →L[ℂ] G)) | |
| (h : Set.centralizer (Set.centralizer (Set.range (diagHom φ ψ)))=Set.range (diagHom φ ψ)) | |
| (v : Intertwiner (BicommutantRange.ofStarHom (diagHom φ ψ) h).commutant) (a : A) : | |
| (component _ v).comp (φ a)=(ψ a).comp (component _ v) := by | |
| apply component_intertwines _ φ ψ v a | |
| have hm : star v.val ∈ ((BicommutantRange.ofStarHom (diagHom φ ψ) h).commutant : Set _) := | |
| (BicommutantRange.ofStarHom (diagHom φ ψ) h).commutant.star_mem' v.mem | |
| rw [VonNeumannAlgebra.coe_commutant] at hm | |
| exact hm _ ⟨a,rfl⟩ | |
| end Kadison.BiprodLinkColumn | |
| end | |
| end | |
| end OAI | |