| import Mathlib | |
| import OAI.Analysis.Commutator.CbDilation | |
| import OAI.Analysis.Commutator.FactorGeneratedSpace | |
| import OAI.Analysis.Commutator.FactorMetric | |
| import OAI.Analysis.Commutator.MetricSimilarity | |
| namespace OAI | |
| noncomputable section | |
| noncomputable section | |
| open scoped CStarAlgebra | |
| namespace Kadison.CbFactor | |
| variable {A H : Type*} [CStarAlgebra A] | |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| local instance cbSimilarityPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A | |
| local instance cbSimilarityStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A | |
| theorem similarity_of_matrix_bound (π : A →ₐ[ℂ] (H →L[ℂ] H)) {c : ℝ} (hc : 0<c) | |
| (hπ : HasMatrixBound π.toLinearMap c) : | |
| ∃ S : (H →L[ℂ] H)ˣ,∀ a : A, | |
| (S : H →L[ℂ] H)*π (star a)*(↑(S⁻¹) : H →L[ℂ] H)= | |
| star ((S : H →L[ℂ] H)*π a*(↑(S⁻¹) : H →L[ℂ] H)) := by | |
| cases subsingleton_or_nontrivial A with | |
| | inl hA => | |
| let := hA | |
| refine ⟨1,?_⟩ | |
| intro a | |
| simp [Subsingleton.elim a 0] | |
| | inr hA => | |
| let := hA | |
| let ρ := representation π.toLinearMap c hc hπ | |
| let V := leftOperator π.toLinearMap c hc hπ | |
| let W := rightOperator π.toLinearMap c hc hπ | |
| have hfactor : ∀ a : A,π a=V.adjoint.comp ((ρ a).comp W) := by | |
| intro a | |
| exact factorization π.toLinearMap c hc hπ a | |
| let L := leftCompression ρ V W | |
| have hLW : L.comp W=1 := leftCompression_right π ρ V W hfactor | |
| have hint : ∀ a : A,(π a).comp L=L.comp (ρ a) := | |
| leftCompression_intertwines π ρ V W hfactor | |
| exact similarity_of_metric π (L.comp L.adjoint) (gram_strictlyPositive L W hLW) | |
| (gram_intertwines π ρ L hint) | |
| end Kadison.CbFactor | |
| end | |
| end | |
| end OAI | |