| import Mathlib |
| import OAI.Analysis.Commutator.AtomicCoefficients |
| import OAI.Analysis.Commutator.AtomicDecomposition |
| import OAI.Analysis.Commutator.MinimalCorner |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| noncomputable section |
|
|
| open scoped ComplexOrder InnerProductSpace |
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| namespace Kadison.AtomicFactor.Decomposition |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| variable {M : VonNeumannAlgebra H} {p : H →L[ℂ] H} (D : Decomposition M p) |
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|
| theorem finite_dimensional [Finite D.I] (hp : IsMinimalProjection M p) : |
| FiniteDimensional ℂ M := by |
| classical |
| let := Fintype.ofFinite D.I |
| let K : Submodule ℂ (H →L[ℂ] H) := Submodule.span ℂ {p} |
| have hcoeff (a : M) (ij : D.I×D.I) : star (D.v ij.1)*(a:H →L[ℂ] H)*D.v ij.2∈K := by |
| have hpa : p*star (D.v ij.1)=star (D.v ij.1) := by |
| simpa only [star_mul,hp.2.1.isSelfAdjoint.star_eq] using congrArg star (D.source ij.1) |
| obtain ⟨z,hz⟩ := corner_scalar M hp |
| (M.mul_mem (M.mul_mem (M.star_mem' (D.mem ij.1)) a.property) (D.mem ij.2)) |
| (by rw [←mul_assoc,←mul_assoc,hpa]) |
| (by rw [mul_assoc,D.source]) |
| rw [hz] |
| exact K.smul_mem z (Submodule.subset_span (Set.mem_singleton p)) |
| let L : M →ₗ[ℂ] ((D.I×D.I) → K) := |
| { toFun := fun a ij => ⟨star (D.v ij.1)*(a:H →L[ℂ] H)*D.v ij.2,hcoeff a ij⟩ |
| map_add' := by |
| intro a b |
| funext ij |
| apply Subtype.ext |
| change star (D.v ij.1)*((a:H →L[ℂ] H)+(b:H →L[ℂ] H))*D.v ij.2= |
| star (D.v ij.1)*(a:H →L[ℂ] H)*D.v ij.2+star (D.v ij.1)*(b:H →L[ℂ] H)*D.v ij.2 |
| rw [mul_add,add_mul] |
| map_smul' := by |
| intro c a |
| funext ij |
| apply Subtype.ext |
| change star (D.v ij.1)*(c • (a:H →L[ℂ] H))*D.v ij.2= |
| c • (star (D.v ij.1)*(a:H →L[ℂ] H)*D.v ij.2) |
| rw [mul_smul_comm,smul_mul_assoc] } |
| apply FiniteDimensional.of_injective L |
| intro a b hab |
| apply Subtype.ext |
| apply D.eq_of_coefficients_eq |
| intro i j |
| exact congrArg (fun f : (D.I×D.I) → K => ((f (i,j)):H →L[ℂ] H)) hab |
|
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| end Kadison.AtomicFactor.Decomposition |
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| end |
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| end |
| end |
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| end OAI |
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|
