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a97875e | 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 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
namespace Kadison.AtomicFactor.Decomposition
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} {p : H →L[ℂ] H} (D : Decomposition M p)
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
end Kadison.AtomicFactor.Decomposition
end
end
end
end OAI
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